classification od knots in lens spaces

UNIVERSITY OF LJUBLJANA

FACULTY OF MATHEMATICS AND PHYSICS

DEPARTMENT OF MATHEMATICS

Boštjan Gabrovšek

Classification of knots in lens spaces

Doctoral thesis

Adviser: dr. Matija Cencelj

Coadviser: dr. Maciej Mroczkowski

Ljubljana, 2013

Abstract

So far knots have been classi�ed up to a certain number of crossings only for a handful of spaces:

the 3-dimensional Euclidean space, the projective space, and the solid torus, the latter being

classi�ed only up to a so-called �ip. In this thesis we append the in�nite family of lens spaces to

this modest list. As a side product, we re�ne the case of the solid torus by providing a complete

classi�cation of knots in it. In both cases we classify knots up to four crossings and up to �ve

crossings with a few exceptions. We also establish which of the knots in the solid torus are

amphichiral. We will see that for each lens space, a subset of prime knots in the solid torus

gives the classi�cation in the lens space. Since there are very few applicable invariants of links in

L(p, q), a necessary condition formaking a classi�cation in these spaces is to develop invariantsof links in L(p, q).�e �rst invariant we introduce is theHOMFLYPT skeinmodule.�eHOMFLYPT skeinmod-

ule has so far only been calculated only for S3 and the solid torus. We show that the HOM-FLYPT skein module of L(p, 1) is a free R-module and we present a basis of this module foreach p > 1.�e second invariant is the Khovanov homology of the Kau�man bracket skein module ofRP3.Khovanov homology, an invariant of links in R3, is a graded homology theory that categori�esthe Jones polynomial in the sense that the graded Euler characteristic of the homology is the

Jones polynomial. Asaeda, Przytycki, and Sikora generalized this construction by de�ning a

double graded homology theory that categori�es the Kau�man bracket skein module of links

in I-bundles over surfaces, except for the surface RP2, where the construction fails due to thestrange behavior of links when projected to the non-orientable RP2. We categorify the missingcase of the twisted I-bundle over RP2, RP2×I ≈ RP3 ∖ {∗}, by rede�ning the di�erential in theKhovanov chain complex in a suitable manner.

�e classi�cation, the calculations of the HOMFLYPT skein modules of the knots, and the cal-

culations of the Kau�man bracket skein modules of the knots are done by a computer program

that is available online at [10].

Math. Subj. Class. (2010): 57M27, 57M25, 57R56.

Keywords: knot, classi�cation, lens space, solid torus, skein module, Kau�man bracket, HOM-

FLYPT, Khovanov homology, categori�cation.

Acknowledgments

Foremost, I would like to express my sincere gratitude to my adviser, Dr. Matija Cencelj, for his

constructive comments, exceptional guidance, caring and patience.

I would like to thankmy coauthor and coadviser Dr. Maciej Mroczkowski for his collaboration:

all the entertaining (and fruitful) discussions on my visits to Poland; his support and guidance

provided over the past years.

My deepest gratitude also goes to Dr. Jože Malešič, in the �rst place for introducing me to

the beautiful world of knot theory and secondly, for believing in me and helping me with my

studies.

To Professor Dušan Repovš for having kind concern and for inviting me to work with his re-

search team and thus giving me the opportunity to start my research.

I am also thankful to Professor PrimožMoravec and Professor Mauro Costantini for providing

help with the calculations of the fundamental groups.

I would also like to thank Miha Nedeljko for his support and talks about my research problems

(and providing me with his computer when mine started to overheat).

To Melanie Sinnhofer for encouragements and all the discussions on topics that "this margin is

to narrow to contain" [Fermat, 1637].

My gratitude also goes to Urša Markovič, for her support, love, and patience even during hard

times of this study.

I also thank my father for his encouragement and con�dence. And above all I would like to

thank mymother for her care and support provided through this journey in every way imagin-

able.

Contents

1 Introduction 11

2 An overview of knot theory in R3 132.1 Basic de�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Knot tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Knot polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Skein modules of 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 �e Kau�man bracket skein module . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 �e HOMFLYPT skein module . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Khovanov homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 �e Khovanov chain complex . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 �e di�erential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.3 �e homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Knots in L(p, q) and their diagrams 293.1 Lens spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Disk diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Reidemeister moves of the disk diagram . . . . . . . . . . . . . . . . . . . 32

3.3 Punctured disk diagrams of the solid torus . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Punctured disk diagrams of L(p, q) . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Reidemeister moves of the punctured disk diagram of L(p, q) . . . . . . 333.5 Arrow diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5.1 Reidemeister moves of the arrow diagram of the torus . . . . . . . . . . . 35

3.6 Arrow diagrams of L(p, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6.1 Reidemeister moves of the arrow diagram of L(p, q) . . . . . . . . . . . . 363.6.2 Transition from punctured disk diagrams to arrow diagrams . . . . . . . 37

3.6.3 Transition from arrow diagrams to punctured disk diagrams . . . . . . . 38

4 �e HOMFLYPT skein module of L(p, 1) 394.1 �e HOMFLYPT skein module of the solid torus . . . . . . . . . . . . . . . . . . 39

4.2 �e construction of H and the main theorem . . . . . . . . . . . . . . . . . . . . . 424.3 Invariance of H under SL moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4 �e case of L(p, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 �e categori�cation of the Kau�man bracket skein module of RP3 495.1 �e Kau�man bracket skein module of RP3 . . . . . . . . . . . . . . . . . . . . . . 495.2 �e chain complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 �e di�erential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7

8 Contents

5.4 �e homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Classi�cation 636.1 Knot notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Reidemeister moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.3 �e classi�cation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3.1 Classifying knots in the solid torus . . . . . . . . . . . . . . . . . . . . . . 66

6.3.2 Classifying knots in L(p, q) . . . . . . . . . . . . . . . . . . . . . . . . . . 696.4 �e results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4.1 Solid torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.4.2 Lens spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7 Conclusion and open questions 77

Appendices 79

A Table of knots in the solid torus 81

B Equivalences of knots in lens spaces 87

C �e HOMFLYPT skein modules 89

D �e Kau�man bracket skein modules 127

Preface

Knot theory plays an important role in the theory of 3-dimensional manifolds. Although it is

considered as a part of geometric topology, it not only appears in many �elds of mathematics,

but also in �elds such as physics, chemistry and biology.

While knots have been studied in various aspects throughout the whole human history, the

theory’s most intriguing results have been obtained over the last three decades.�e importance

of the theory can perhaps be demonstrated by the fact that four mathematicians have already

received Fields medals for their results in this theory: V. Jones, E. Witten, V. G. Drinfeld, and

M. L. Kontsevich.

In this thesis, we study the theory of knots from the perspective of geometric topology with the

chapters grouped into three parts:

InChapter 2 andChapter 3we overview the existing theory of knots and links in 3-manifolds.

In Chapter 4 and Chapter 5 we develop new invariants of lens spaces, namely, the HOMFLYPT

skein module of lens spaces L(p, 1) (and conjecture about L(p, q), q > 1) [14] and Khovanovhomology of the Kau�man bracket skein module of RP3 ≈ L(2, 1) [11].Chapter 6 is devoted to the classi�cation of knots in the solid torus and in L(p, q). As a productwe produce knot tables that are presented in the appendices.

Section 3.6 and Chapter 4 is the result of the joint work with the author’s coadviser dr. Maciej

Mroczkowski [13, 14]. Chapter 6 is the result of the joint work with the author’s coadviser (clas-

si�cation of knots in the solid torus up to a �ip) [12] and was generalized by the author for the

purpose of this paper (classi�cation of knots in L(p, q)).�e computer algorithm presented inChapter 6 was primarily written by the coadviser and the author, for the purpose of calculating

the Kau�man bracket skein module of knots and the classi�cation of knots in the solid torus

(up to �ips) [12], and was rewritten by the author for the calculations of the HOMFLYPT skein

modules of links and the classi�cation of links in the solid torus and L(p, q).

9

1Introduction

From the mathematical perspective, knots were �rst mentioned in a 1771 paper by A. T. Van-

dermonde [56], where braids and knots are speci�cally placed as a subject of the geometry of

position.

�e �rst notable knot invariant, the Gauss linking integral, was introduced by C. F. Gauss in

1833. Although the construction was inspired by astronomy, the result is nowadays known as

the linking number. It was not until 1885 when P. G. Tait developed the �rst knot table [52].�e

motivation behind such a classi�cation was Lord Kelvin’s (apparently mistaken) theory that

atoms are knots in the aether.

Following the advancement of topology, knot theory became a widespread �eld of study at the

beginning of the 20th century with early pioneers ofmodern knot theory beingK. Reidemeister,

J. W. Alexander, M. Dehn, and R. Fox, to name a few. In 1984 V. Jones furthermore popularized

the subject with the discovery of the Jones polynomial [28], which led to the discovery of other

polynomials, such as the HOMFLYPT polynomial and various Kau�man polynomials.

Another major breakthrough in the study of knots appeared in the late 1990s by the seminal

work of M. Khovanov, who managed to construct a homological theory that generalizes the

Jones polynomial. Similarly, in 2002 P. Ozsváth and Z. Szabó generalized the Alexander poly-

nomial by introducing knot Floer homology [40].

A systematic study of knots in spaces other than the 3-dimensional Euclidean space started in

1987 when Przytycki and Turaev introduced the study of skein modules [44, 55] and continued

in 1991 with Yu. V. Drobotukhina’s classi�cation of knots in the projective space [17]. Since

then, a considerable part of contemporary knot theory has been devoted to the study of knots

in various 3-dimensional spaces [46].

11

2An overview of knot theory in R3

2.1 Basic definitions

Standard terminology, as well as the de�nitions of the polynomials in the next section can be

found in [1, 7, 34].

Knot theory is inspired by what we perceive as a "knot" in our daily life: an entangled piece

of string; although the theory is for the most part interested in "knots" that are connected, i.e.

joined together at the ends.

A (mathematical) knot is an embedding of the circle S1 in the 3-dimensional Euclidean spaceR3 or in the 3-sphere S3. A link is a embedding of a disjoint union of n circles intoR3 (S3).

Studying the embeddings of circles has little in common of the intuitive idea of what a knot

should be. �e topological object that we will be interested in, is a class of embeddings which

more naturally corresponds to the ideas portrayed above. We �rst set up some de�nitions.

Two embeddings f0, f1 ∶ X → Y are isotopic if they can be connected through embeddings,more precisely, if there exists a homotopy

H ∶ X × I Ð→ Y ,

such that H(x , 0) = f0(x), H(x , 1) = f1(x) and for each t ∈ I, H(−, t) is an embedding.We see that isotopy still does not apply to our situation, since any two knots can be shown

to be isotopic: any "knotted" areas of a knot can be continuously contracted to a point (see

Figure 2.1) [7].

Ð→ Ð→ Ð→

Figure 2.1: Contracting a "knotted" area to a point.

What we wish to achieve is, not only to isotope the embedding, but carry the whole ambient

space along with us.�is is achieved through ambient isotopy.

13

14 2.1. Basic de�nitions

Two embeddings f0, f1 ∶ X → Y are ambient isotopic, if there exists a homeomorphism (ordi�eomorphism in the smooth category) H ∶ Y → Y which is isotopic to the identity map IdY ,with the property that f1 = H ○ f0.

In the smooth category, ambient isotopy and isotopy coincide [22]. Rather than carrying the

adjective "ambient" throughout the text, we will, from now on, work only in the category of

smoothmanifolds.�is also eliminates any pathological behavior of knots, such as the existence

of wild knots (Figure 2.2).

Figure 2.2: An example of a wild knot.

An isotopy forms an equivalence relation on the set of knots and we will o�en use the word

"knot" when referring to a knot’s isotopy class. A knot invariant is a function on the knot thatis invariant under isotopy.

To have a working theory of knots, we must introduce suitable knot diagrams.

Let p be a projection of the knot K to a plane Σ ⊂ R3. We require p to satisfy the followingproperties (see Figure 2.3):

• no three points project to the same point,

• all double points meet transversely,

• no cusps occur.

(a) triple point (b) self tangency (c) a cusp

Figure 2.3: Forbidden projections in a regular diagram.

A projection satisfying above conditions is called a regular projection.�roughout this thesiswe will consider every projection to be regular.

Double points in a projection are called crossings. �e edges of a knot’s projection are calledarcs and faces are called regions.�e shadow of a knot is the image of the (regular) projectionof a knot (Figure 2.4(a)), a knot diagram is the image containing the additional informationwhether the crossing is an over- or undercrossing (Figure 2.4(b)). A knot can be reconstructed

from its diagram up to isotopy.

A tangle is the embedding of a disjoint union of arcs and circles into the 3-ball I3, where theembedding sends the endpoints of the arcs to the boundary of I3. A n-tangle is a tangle con-sisting of n arcs (and possibly some circles) where n endpoints lie in {0} × I2 and n endpointslie in {1} × I2 (Figure 2.4(c)) [8].

Chapter 2. An overview of knot theory in R3 15

(a) a shadow (b) a knot diagram (c) a 2-tangle

Figure 2.4: A shadow, a knot diagram and a tangle.

Knot diagrams can be transformed to other diagrams using Reidemeister moves R-I, R-II, and

R-III presented in Figure 2.5 [49]. Each move is a local transformation on a small region of the

diagram. For these moves the following Reidemeister theorem holds:

←→

(a) R-I

←→

(b) R-II

←→

(c) R-III

Figure 2.5: Reidemeister moves.

�eorem 2.1.1 (Reidemeister [49]). Two diagrams represent equivalent knots if and only if thereexists a �nite sequence of Reidemeister moves R-I, R-II, and R-III that transform one diagram tothe other.

�e proof is quite elementary in nature: we have to study what happens with the knot’s projec-

tion when we isotope the knot in a way that in one step of the isotopy a forbidden singularity

from Figure 2.3 appears in the projection.

A �ype1 is a local transformation of a knot diagram presented in Figure 2.6. A �ype is geomet-rically interpreted by �ipping a 2-tangle.

T ∼ T

Figure 2.6: A �ype.

�e knot that allows a diagram with no crossings is called the unknot O (Figure 2.7(a)), al-though sometimes we will refer to it as the trivial knot. A kink is the loop created by R-I (seethe right-hand side of Figure 2.5(a)).

If a knot has been given an orientation, we call such a knot an oriented knot. In a diagram,the orientation is indicated by one or more arrows (Figure 2.7(b)). When talking about isotopy

between oriented knots, we require the isotopy to preserve the orientation of the knot.

1An old Scottish word meaning to turn or to fold back [18].

16 2.1. Basic de�nitions

(a) the unknot (b) an oriented knot

Figure 2.7:�e unknot and an oriented knot.

If we reverse the orientation of an oriented knot K, the resulting knot is called the reverse of Kand is denoted by −K (Figure 2.8).

(a) K (b) −K

Figure 2.8:�e reverse of a knot.

In a diagram of an oriented knot, each crossing can be assigned an integer +1 or −1, accordingto the right-hand rule pictured in Figure 2.9.�e sum of all the crossing signs of a diagram Dis called the writhe wr(D) of D.

(a) positive crossing (b) negative crossing

Figure 2.9:�e sign of a crossing.

�eminimal number of crossings a knot K can have in any of its diagrams is called the crossingnumber cr(K) of K.

�e connected sum of two oriented knots K1 and K2 is formed by removing a small arc fromeach knot and connecting the four endpoints by two new arcs bounding a band in such a way

that the orientation stays consistent with the original knots, the result being a single (oriented)

knot K1#K2 (Figure 2.10).

K1 K2

Ð→

K1 #K2Figure 2.10:�e construction of a connected sum of two knots.


We call a knot prime if it cannot be written as the connected sum of two non-trivial knotsand composite otherwise. In the case of unoriented knots, the connected sum in not well-de�ned: in general there are two distinct equivalence classes of the resulting knot.�e equiva-

lence class depends on the relative orientations chosen in order to perform the connected sum

operation.

�e following theorem regarding connected sums was shown by Schubert [51]:

Lemma 2.1.1 (Unique decomposition of knots). Every knot can be uniquely decomposed, up tothe order in which the decomposition is performed, as the connected sum of prime knots.

If we embed a ribbon S1× I inR3 instead of a circle, such an embedding is called a framed knot(Figure 2.11). One o�en represents diagrams of framed knots with unframed diagrams, in this

case, blackboard framing is assumed. Blackboard framing corresponds to a ribbon lying �at onthe projection plane. In case the knot has a di�erent framing than the blackboard one, we specify

the (relative) framing number, i.e. additional le� or right twists. If we assume the ribbon S1 × Iis oriented, we speak of oriented framed knots. We present such knots with ordinary oriented

knot diagrams, where blackboard framing is again assumed and the orientation is induced by

K × {0}.

Figure 2.11: A framed knot.

�eorem2.1.2 (Kau�man [30]). Twodiagrams of framed knots represent equivalent framed knotsif and only if there exits a �nite sequence of Reidemeister moves R-II and R-III that transform onediagram to the other.

Figure 2.12 demonstrates that a framed knot is not invariant under R-I.

∼

Figure 2.12: Removal of a kink changes the framing of a knot.

�e equivalence relation of knot diagrams generated by only Reidemeister moves R-II and R-III

is called regular isotopy.

If we re�ect a knot K through a plane, the re�ection is called the mirror of K and is denotedby K. To construct a mirror knot from a knot diagram, we exchange all overcrossings withundercrossings and vice versa (Figure 2.13). A knot is amphichiral if it is equivalent to itsmirror image and chiral if it is not.

Let us for a moment assume, that a knot K lies in an arbitrary 3-manifold M. We call a knota�ne, if it lies inside a 3-ball B3 ⊂ M and non-a�ne otherwise. Note that all knots in S3 area�ne.

18 2.2. Knot polynomials

(a) K (b) K

Figure 2.13:�e mirror of a knot.

2.1.1 Knot tables

One of the earliest motivations in knot theory was the classi�cation of knots. Such a classi�-

cation is usually presented by providing a knot table. In such a table, we denote each knot bya symbol nk, where n presents the crossing number and k represents the index of the knot inthe table among knots with n crossings (i.e. nk lies in the k-th place among n-crossing knots).If, for a given 3-manifold, the mirror operation is well-de�ned, we only put one of the possible

pair in the table (and preferably provide information if the knot is amphichiral or not). If the

operation of connected sums is well-de�ned in the manifold, we do not, due to Lemma 2.1.1,

include connected sums in a knot table. Similarly, a classi�cation of links is given by a linktable.

�ere are various knot and link tables of knots and links in S3, but the most widely used con-vention is to use the Rolfsen knot table for knots and the�istlethwaite link table for links, both

of which can be found at [5].

As an example, the �rst few knots in the Rolfsen knot table are presented in Table 2.1.

Table 2.1: Table of knots in S3 up to 5 crossings.

01 31 41 51 52

For the case of classifying knots in 3-manifolds that are not S3, a knot table for knots inRP3 canbe found in [17] and a partial classi�cation of knots in the solid torus can be found in [12].

2.2 Knot polynomials

A knot polynomial is a polynomial that is a knot invariant. �e �rst knot polynomial, theAlexander polynomial, was discovered by J. W. Alexander II in 1923 [2] and the second one was

the Jones polynomial discovered in the early 1980s by V. Jones [28]. Jones’s discovery quickly

led to an outburst of various other polynomials, most notably the Conway’s bracket polynomial,

the Kau�man polynomial, and the HOMFLYPT polynomial [1, 34].


We outline the de�nitions of some of the most commonly used polynomials that we will refer

to throughout the thesis.

Let L+, L−, and L0 be oriented links that are identical except inside a small 3-ball, where theirprojections look like those presented in Figure 2.14. Such a triple of links is referred to as the

skein triple and a relation involving such a triple is called a skein relation.

(a) L+ (b) L− (c) L0

Figure 2.14:�e skein triple.

�e normalized Alexander polynomial2 ∆(L) of an oriented link L is a Laurent polynomialin t 1⁄2 and is characterized by the following conditions:

∆(O) = 1, (normalization)

∆(L+) − ∆(L−) = (t 1⁄2 − t−1⁄2)∆(L0), (skein relation)

with O being the unknot.

�e Alexander-Conway polynomial ∇(L) is a polynomial in z and is characterized by:∇(O) = 1, (normalization)

∇(L+) −∇(L−) = z∇(L0). (skein relation)

�e Jones polynomial J(L) of an oriented link L is a Laurent polynomial in t 1⁄2 and is de�nedby the following relations:

J(O) = 1, (normalization)

t−1J(L+) − tJ(L−) = (t 1⁄2 − t−1⁄2)J(L0). (skein relation)

�eKau�manbracket3 ⟨ q⟩ is an unoriented framed link polynomial invariant in variableA [31].�e bracket is de�ned by the state-sum formula, for which we �rst need to set up a few de�ni-

tions.

LetD be an oriented link diagramwith n crossings labeled arbitrarily from 1 to n and denote thisset of crossings by X .�e number of positive crossings in D is marked by n+ and the numberof negative crossings is marked by n−.

Each crossing can be smoothened by a smoothening of type 0 or 1 according to Figure 2.15.

We call {0, 1}X the discrete cube of D and a vertex s ∈ {0, 1}X a (Kau�man) state of D. Eachstate corresponds to a diagram with each crossing smoothened either by a type 0 or a type 1

smoothening. For convenience, this complete smoothening is also called a state of D. Eachstate is just a collection of disjoint closed loops which are called circles, the number of circlesin a state s is denoted by ∣s∣.

2Alexander’s original polynomial is unique only up tomultiplication by the Laurentmonomial±tn/2, so usuallyone �xes a certain normalization, which may not necessarily be the one used in our characterization.

3Also known as the Bracket polynomial.

20 2.2. Knot polynomials

Ð→

(a) type 0 smoothening

Ð→

(b) type 1 smoothening

Figure 2.15: Two types of smoothenings.

By #0(s)we denote the number of 0 factors in s and by #1(s) the number of 1 factors in s.For a given diagram D of the link L, the Kau�man bracket ⟨D⟩ is de�ned by the state-sumformula:

⟨D⟩ = ∑s∈{0,1}X

A#0(s)−#1(s)(−A2 − A−2)∣s∣−1⟨O⟩,

where we use the evaluation ⟨O⟩ = 1. A non-standard version of the Kau�man bracket (that willbe used in Chapter 5) uses the evaluation ⟨∅⟩ = 1 and can be expressed by the formula:

⟨D⟩ = ∑s∈{0,1}X

A#0(s)−#1(s)(−A2 − A−2)∣s∣⟨∅⟩.

As we have already stated, the Kau�man bracket is an invariant of framed links, since it is in-

variant only under regular isotopy, i.e. not invariant under R-I [30].

We continue by presenting a recursive characterization of ⟨ q⟩.Let L, L0, and L∞ be unoriented links that are identical except inside a small 3-ball, where theirprojections look like those presented in Figure 2.16. We will refer to such a triple as aKau�mantriple and a relation involving this triple as a Kau�man relation.

(a) L (b) L0 (c) L∞

Figure 2.16:�e Kau�man triple.

�e Kau�man bracket is characterized by the following relations [30]:

⟨L⟩ = A⟨L0⟩ + A−1⟨L∞⟩, (Kau�man relation)

⟨L ⊔ O⟩ = (−A2 − A−2)⟨L⟩, (framing relation)

⟨O⟩ = 1. (normalization)

Proposition 2.2.1. For the connected sum of knots K and K′, the following equality holds:

⟨K#K′⟩ = ⟨K⟩⟨K′⟩.

Proof. We prove this by induction n, where n is the number of crossings of K′. For n = 0 theequality ⟨K#O⟩ = ⟨K⟩⟨O⟩ holds, since K#O = K and ⟨O⟩ = 1. For n > 0 we resolve a crossing ofK′ using the Kau�man relation:

⟨K#K′⟩ = A⟨K#K′0⟩ + A−1⟨K#K′

∞⟩.


Since K′0 and K′

∞ both have n − 1 crossings, we use the induction hypothesis:

⟨K#K′⟩ = A⟨K⟩⟨K′0⟩ + A−1⟨K⟩⟨K′

∞⟩ = ⟨K⟩(A⟨K′0⟩ + A−1⟨K′

∞⟩) = ⟨K⟩⟨K′⟩,

where the last equality holds by the Kau�man relation on the previously resolved crossing.

If, for a given diagram, wemultiply the bracket polynomial by (−A3)−wr(L), we get theKau�manpolynomial X:

X(L) = (−A3)−wr(L)⟨L⟩,which, due to the normalization, becomes an invariant under isotopy and is therefore an invari-

ant of unframed links [31].

�e polynomial X equals the Jones polynomial by a change of variable [1]:

X(L)(A) = J(L)(A−4).

Since it holds that wr(K#K′) = wr(K) + wr(K′), the formula X(K#K′) = X(K)X(K′) alsoholds for X and thus for the Jones polynomial:

J(K#K′) = J(K)J(K′).

At this point, we de�ne a variant of the Jones polynomial that will be used in the construction

of the Khovanov homology in Section 2.4. Khovanov uses a slightly modi�ed version of the

Kau�man bracket that is de�ned by the following axioms [32]:

⟨L⟩m = ⟨L0⟩m − q⟨L∞⟩m , (Kau�man relation)

⟨L ⊔ O⟩m = (q + q−1)⟨L⟩m , (framing relation)

⟨∅⟩m = 1. (normalization)

To turn this bracket into a link invariant, we must multiply it by (−1)n−qn+−2n− .�is leads us tothe unnormalized Jones polynomial J:

J(L) = (−1)n−qn+−2n−⟨L⟩m .

�e Laurent polynomial J in variable q is an invariant of oriented links [50].

�eHOMFLYPT4 polynomial of an oriented knot is a 2-variable Laurent polynomial in vari-ables v and z de�ned by the following relations5:

P(O) = 1, (normalization)

v−1P(L+) − vP(L−) = zP(L0). (skein relation)

4�e HOMFLYPT is o�en referred to as the HOMFLY polynomial. It was independently discovered by �ve

di�erent groups: Hoste; Ocneanu; Millett and Lickorish; Freyd and Yetter; Przytycki and Traczyk. Unfortunately,

because of the martial law in Poland, the discovery of Przytycki and Traczyk did not reach the United States before

D. Yetter coined the original acronym [27].5Due to various authors, about four distinct de�nitions of theHOMFLYPTpolynomial can be found inmodern

literature, each involving di�erent signs and letters.

22 2.3. Skein modules of 3-manifolds

To demonstrate the strength of the HOMFLYPT polynomial, we note that all of the previously

de�ned polynomials can be obtained from the HOMFLYPT polynomial [1, 50]:

∆(t) = P(1, t 1⁄2 − t−1⁄2),∇(z) = P(1, z),J(t) = P(t, t 1⁄2 − t−1⁄2),

X(A) = P(A−4,A−2 − A2),⟨ q⟩ = (−A3)w( q)P(A−4,A−2 − A2).

2.3 Skein modules of 3-manifolds

Skein modules have their origin in the observation made by J. W. Alexander that the three

polynomials ∆(L+), ∆(L−), and ∆(L0) of links L+, L−, and L0, respectively, are linearly relatedby the skein relation6

∆(L+) − ∆(L−) = (t 1⁄2 − t−1⁄2)∆(L0).

J. H. Conway pursued this idea by considering the free Z[z]-module over the set of isotopyclasses of links in S3modulo theZ[z]-module generated by the skein relation of the Alexander-Conway polynomial [42, 46].

By formalizing such a construction and generalizing it for any 3-manifold (not just S3), J. H.Przytycki andV.GTuraev introduced the theory of skeinmodules in 1987 [55, 44]. In Przytycki’s

own words, the theory of skein modules is the idea of building an algebraic topology based on

knots.�e building blocks of such a theory are considered up to isotopy, where instead of formal

linear combinations of simplices we use linear combinations of links and instead of boundary

relations, we use properly chosen skein relations [42].

In the remaining part of this section we overview the theory of skein modules as described

in [46, 44].

LetM be a 3-manifold and let R be a commutative ring with identity. LetL(M) be the set of iso-topy classes of links inM and let RL(M) the free R-module generated by L(M). Let S(M;R)be the submodule generated by a collectionR ⊂ RL(M) of �nite formal expressions

r0L0 + r1L1 +⋯ + rL∞,

where r, r0, r1, r2, . . . ∈ R and L0, L1, L2, . . . being classes of links that are identical except in theparts shown in Figure 2.17.

�e skein module S(M;R,R) is RL(M)modulo S(M;R):

S(M;R,R) = RL(M)/S(M ,R).

Example 2.3.1. S(M;R;∅) = RL(M).6�e Alexander polynomial was originally de�ned through certain manipulation of diagrams [2].


(a) L0 (b) L1 (c) L2 (d) L3

⋯

(e) L∞

Figure 2.17:�e skein tuple.

Example 2.3.2. S±(M;R) = S(M;R, − ) is a free R-module over the homotopy classes ofclosed curves in M (i.e. two links L and L′ are equivalent in S±(M) if they are homotopic) [46].�e relation − means that we identify all links that are identical except inside a 3-ball whereone of them looks like and the other one looks like .

We continue by describing two particular skein modules in detail: the Kau�man bracket skein

module and the HOMFLYPT skein module.

2.3.1 The Kauffman bracket skein module

�e Kau�man bracket skeinmodule (KBSM) generalizes the Kau�man bracket and is one of the

most extensively studied skein module [39] as it has been calculated for a number of di�erent

3-manifolds [43, 26, 38, 39].

To construct KBSM of a 3-manifold M, take a coe�cient ring R with A ∈ R being a unit (anelement with a multiplicative inverse). Since, as in the case of the Kau�man bracket, we would

like to study framed links, we let Lfr(M) be the set of isotopy classes of framed links in M,including the class of the empty link [∅]. As before, let RLfr(M) be the free R-module spannedby Lfr(M). We assume the ring R is �xed and omit it in further notations.

Wewould like to impose the Kau�man relation and the framing relation in RLfr(M). We there-fore take the submodule Sfr(M) of RLfr(M) generated by

− A − A−1 , (Kau�man relator)

L ⊔ − (−A2 − A−2)L. (framing relator)

�e Kau�man bracket skein module S2,∞(M) is RLfr(M)modulo these two relations:

S2,∞(M) = RLfr(M)/S(M).

Example 2.3.3. For the 3-sphere, S2,∞(S3) is a free R-module with the basis being just the equiva-lence class of the unknot. Expressing a knot in this basis and evaluating [O] = 1, we get exactly theKau�man bracket. Similarly, we can set the basis to be the empty knot and by evaluating [∅] = 1,we get the previously described non-standard version of ⟨ q⟩ (these two facts can be easily seen asthe de�nition corresponds directly to the de�nition of the polynomial).

24 2.3. Skein modules of 3-manifolds

�e Kau�man bracket skein module of the solid torus T has been calculated in [55]:

�eorem 2.3.4. KBSM of the solid torus T is freely generated by an in�nite set of generators{xn}∞n=0, where xn, n > 0, is a parallel copy of n longitudes of T and x0 is the a�ne unknot.

�e Kau�man bracket skein module of L(p, q) has been calculated in [24]:

�eorem 2.3.5. KBSM of L(p, q) is freely generated by the set of generators {xn}⌊p/2⌋n=0 , where xn,

n > 0, is a parallel copy of n longitudes of T ⊂ L(p, q) and x0 is the a�ne unknot (see Chapter 3).

2.3.2 The HOMFLYPT skein module

�eHOMFLYPT skeinmodule of a 3-manifoldM generalizes theHOMFLYPTpolynomial. Letthe ring R this time have two units v , z ∈ R. Let Lor(M) be the set of isotopy classes of orientedlinks in M, including the class of the empty link [∅] and let RLor(M) be the free R-modulespanned by Lor(M). We again assume R is �xed and omit it in the notations.

As in the case of the HOMFLYPT polynomial, we would like to impose the HOMFLYPT skein

relation in RLor(M). We take the submodule S(M) of RLor(M) generated by the expres-sions

v−1 − v − z . (HOMFLYPT relator)

We also add to S(M) the HOMFLYPT relation involving the empty knot:

v−1∅ − v∅ − zO . (HOMFLYPT relator)

�eHOMFLYPT skein module S3(M) ofM is RLor(M)modulo the above relations:

S3(M) = RL(M)/S(M).

Example 2.3.6. For the 3-sphere S3, S3(S3) is freely generated by [O]. By expressing a knot K inthis basis and evaluating [O] = 1, we get the HOMFLYPT polynomial of K.

Example 2.3.7. �e HOMFLYPT skein module of the solid torus T is a free R-module, generatedby an in�nite set of generators (see Chapter 4 for details on the basis of this module) [23, 55].

Remark. �e integer subscript in the skein module indicates what types of links are used in therelations of the module. A skein module denoted by Sk(M) indicates the relations are gener-ated by skein expressions r0L0 + r1L1 + ⋯rk−1Lk−1, with r0, r1, . . . , rk−1 ∈ R and L0, L1, . . . , Lk−1being classes of links from Figure 2.17. Similarly, Sk,∞(M) indicates that we are also taking intoaccount L∞, i.e. the module is generated by expressions r0L0 + r1L1 + ⋯rk−1Lk−1 + rL∞, withr, r0, r1, . . . , rk−1 ∈ R.


2.4 Khovanov homology

Amajor breakthrough in the study of knots in S3 appeared in the late 1990s by a series of lecturesby M. Khovanov, who managed to categorify the Jones polynomial by constructing a chain

complex of graded vector spaces with the property that the homology of this chain complex,

the Khovanov homology, is a link invariant. Moreover, the graded Euler characteristic of this

complex is the Jones polynomial [32]. Perhaps themost outstanding consequence of this theory

is the s-invariant of Rasmussen, which gives a bound on a knot’s slice genus and is su�cient toprove the Milnor conjecture [47]. More recently, Kronheimer and Mrowka showed that the

Khovanov homology detects the unknot [33].

Khovanov’s categori�cation is the idea to replace polynomials with graded vector spaces of

appropriate graded dimensions to turn the Jones polynomial into a homological object. �e

Kau�man bracket ⟨ q⟩ is replaced with the Khovanov bracket J qK, which is a chain complex ofgraded vector spaces whose graded Euler characteristic, as we have mentioned before, is the

Jones polynomial [4].

Since wewill use the construction of the Khovanov chain complex (although substantiallymod-

i�ed) in Chapter 5, we at this point repeat the classical construction of the Khovanov homology

as it is constructed in [4] verbatim. �e Khovanov chain complex and the Khovanov homol-

ogy will be constructed using graded vector spaces (as it is done in [4]), but the construction

works just as well, with suitable modi�cations, overZ-modules – such a construction is used tocategorify KBSM of RP3 in Chapter 5.

2.4.1 The Khovanov chain complex

Let W = ⊕m Wm be a graded vector space with homogeneous components {Wm}m. �egraded dimension ofW is the power series

q-dimW =∑mqm dimWm .

Let q {l} be the degree shi� operation on graded vector spaces. �at is, if W = ⊕m Wm is a

graded vector space we setW{l}m =Wm−l , so that q-dimW{l} = qlq-dimW .

Likewise, let q [s] be the height shi� operation on chain complexes. �at is, if C is a chaincomplex

⋯Ð→ Cr−1 Ð→ Cr Ð→ Cr+1 Ð→ ⋯

of graded vector spaces and if C = C[s], then Cr = Cr−s (with di�erentials shi�ed accord-ingly).

Let X , n and n± be as section 2.2. Let V be the graded vector space with two basis elements v±whose degrees are ±1, respectively. Note that q-dimV = q + q−1. With every state s ∈ {0, 1}X ofthe discrete cube {0, 1}X we associate the graded vector space V(L) = V⊗∣s∣{r}, with ∣s∣ againbeing the number of circles in s. We then set the r-th chain group JLKr (for 0 ≤ r ≤ n) to be the

26 2.4. Khovanov homology

direct sum of all the vector spaces associated with the states with the number of 1-smoothenings

being r:JLKr = ⊕

s∶r=#1(s)Vs(L).

Ignoring that JLK is not yet a complex, for we have not yet endowed it with a di�erential, we setC(L) = JLK[−n−]{n+ − 2n−}.

�e graded Euler characteristic χq(C) of a chain complex C is de�ned to be the alternating sumof the graded dimensions of its homology groups, and, if the degree of the di�erential d is 0and all chain groups are �nite dimensional, it is also equal to the alternating sum of the graded

dimensions of the chain groups. A few paragraphs below we will endow C(L) with a degree 0di�erential.�is granted and given that the chains of C(L) are already de�ned, we can state thefollowing theorem:

�eorem 2.4.1 ([4]). �e graded Euler characteristic of C(L) is the unnormalized Jones polyno-mial of L:

χq C(L) = J(L).

Next, we wish to turn the sequence of spaces C(L) into a chain complex.

2.4.2 The differential

�e edges of the discrete cube {0, 1}X can be labeled by sequences in {0, 1, ⋆}X with just one⋆ (so the tail of such an edge is found by setting ⋆ → 0 and the head by setting ⋆ → 1). �enumber #1(ξ) of an edge ξ is de�ned to be the number of 1’s in its tail, and hence if the maps onthe edges are called dξ, then we collapse the maps with the same number of 1’s:

dr = ∑#1(ξ)=r

(−1)ξdξ .

It remains to explain the signs (−1)ξ and to de�ne the per-edge maps dξ. �e former is easy.

To get the di�erential d to satisfy d ○ d = 0, it is enough that all square faces of the cube wouldanti-commute. But it is easier to arrange the dξ’s so that these faces would (positively) com-

mute; so we do that and then change signs to make the faces anti-commutative.�is is done by

multiplying dξ by (−1)ξ = (−1)∑i< j ξ i , where j is the location of the ⋆ in ξ.

It remains to �nd maps dξ that make the cube commutative (when taken with no signs) and

that are of degree 0.�e space Vs on each state s has as many tensor factors as there are circlesin the smoothing of corresponding to s.�us we put these tensor factors in Vs and circles in sin bijective correspondence. Now for any edge ξ, the smoothing at the tail of ξ di�ers from thesmoothing at the head of ξ by just a little: either two of the circles merge into one or one of thecircle splits in two. So for any ξ, we set dξ to be the identity on the tensor factors corresponding

to the circle that do not participate, and then we complete the de�nition of dξ using two linear

maps m ∶ V ⊗ V → V and ∆ ∶ V → V ⊗ V as follows:


( ÐÐ→ )Ð→ (V ⊗ V mÐÐ→ V) m ∶

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

v+ ⊗ v− ↦ v−v− ⊗ v+ ↦ v−v+ ⊗ v+ ↦ v+v− ⊗ v− ↦ 0

( ÐÐ→ )Ð→ (V ∆ÐÐ→ V ⊗ V) ∆ ∶ { v+ ↦ v+ ⊗ v− + v− ⊗ v+v− ↦ v− ⊗ v−

Note that because of the degree shi�s in the de�nition of Vs and because we want dξ to be of

degree 0, the maps m and ∆ must be of degree −1. Also, as there is no canonical order on thecircles in s (and hence on the tensor factors of Vs), m and ∆ must be commutative and co-commutative respectively.�ese requirements force the equalitym(v−⊗ v−) = m(v−⊗ v+) andforce the values of m and ∆ to be as shown above up to scalars.

2.4.3 The homology

We conclude with the following proposition and theorem.

Proposition 2.4.1 (Khovanov [32]). �e sequences JLK and C(L) are chain complexes.

LetHr(L) denote the r-th cohomology of the complex C(L). It is a graded vector space depend-ing on the link projection L. Let P(L) denote the graded Poincaré polynomial of the complexC(L) in the variable t:

P(L) =∑rtrq-dimHr(L).

Proposition 2.4.2 (Khovanov [32]). �e graded dimensions of the homology groups Hr(L) arelink invariants, and hence P(L), a polynomial in the variables t and q, is a link invariant thatspecializes to the unnormalized Jones polynomial at t = −1.

3Knots in L(p, q) and their diagrams

In 1884W. Dyck introduced the concept of constructing a lens space, but unfortunately did not

publish more than just ideas before withdrawing from topology completely.�e �rst intensive

study of lens spaces was done by H. Tieze in 1908, who laid down the importance of such spaces

since they are simple on the one hand, but have intriguing properties and diverse applicability

on the other hand [54]. �e term "linsenräume" (lens space) itself was coined by Seifert and

�relfall in 1930 [21].

In her seminal work, Yu. V. Drobotukhina introduced the study of knots in the real projective

space. She constructed a version of the Jones polynomial for RP3 [16] and managed to classifylinks in RP3 up to 6 crossings [17]. J. Hoste and J. H. Przytycki furthermore generalized theJones polynomial by calculating KBSM of L(p, q)[24].

In order to have a working theory of links in L(p, q), we start o� by de�ning a lens space andcontinue by de�ning a diagram of a link in it.

3.1 Lens spaces

At least �ve more or less distinct de�nitions of the 3-dimensional lens space can be found in

modern topology textbooks [57]. Out of these �ve, we overview three models, out of which the

second and thirdmodel will be used throughout the rest of the thesis. For all three constructions

we assume 0 < q < p are two coprime integers.

First model of L(p, q). Let S3 = {(z0, z1) ∈ C2 ∶ ∣z0∣2 + ∣z1∣2 = 1} be the 3-sphere.�e lens spaceis de�ned to be the orbit space of the free action of the cyclic group Z/p on S3 given by

n q(z0, z1)z→ (z0 qe2πin/p, z1 qe2πiqn/p).Second model of L(p, q). Consider the 3-ball B3 = {(x , y, z) ∈ R3 ∶ x2 + y2 + z2 ≤ 1} withS+ = ∂B3 ∩ {(x , y, z) ∈ R3 ∶ z ≥ 0} and S− = ∂B3 ∩ {(x , y, z) ∈ R3 ∶ z ≤ 0} being the upper andlower closed hemispheres of ∂B3, respectively. Let rp,q ∶ S+ → S+ be the rotation of S+ by 2πq/paround the z-axis and letm ∶ S+ → S− be the re�ection with respect to the xy-plane. We identifyeach pointw ∈ S+ on the upper hemisphere with the pointm ○ rp,q(w) on the lower hemisphere

29

30 3.1. Lens spaces

(Figure 3.1).�e lens space L(p, q) is the quotient of B3 by this equivalence relation:

L(p, q) = B3/ ∼ .

Note that on the equator {(x , y, 0) ∈ R3, x2+y2 = 1} each equivalence class contains p points.

w

m ○ rp,q(w)

π4/7

Figure 3.1: Representation of L(7, 2).

�ird model of L(p, q). Take a solid torus T = S1 ×D2 and �x a point x0 ∈ ∂T on its boundary.Let l andm be the generic longitude and meridian of T as shown in Figure 3.2. Note that l andm generate the fundamental group π1(∂T , x0).

l m

Figure 3.2:�e generic longitude l and meridian m of T .

If a curve on ∂T represents the class p[l]+q[m] ∈ π1(T , x0), we call such a curve a (p, q)-curve.�e lens space L(p, q) is the result of gluing to solid tori T1 and T2 along their boundaries viathe homeomorphism hp,q ∶ ∂T1 → ∂T2 that takes a meridian m ⊂ ∂T1 to a (p, q)-curve in ∂T2(see Figure 3.3 for an example of p = 3, q = 1).

m

h3,1ÐÐ→

h3,1(m) = (3, 1)Figure 3.3:�e boundary homeomorphism h3,1 ∶ ∂T1 → ∂T2.

Proposition 3.1.1. All three models agree.

For this folklore fact, a comprehensive proof can be found in [57].

�e projective space1 RP3 is the lens space L(2, 1).1�e 3-dimensional real projective space is de�ned to be the set of lines in R3 through the origin. Using ho-

mogeneous coordinates it is not hard to show the validity of the statement.

Chapter 3. Knots in L(p, q) and their diagrams 31

�e classi�cation problem of 3-dimensional lens spaces up to homeomorphism was solved by

Reidemeister in 1935 [48]:

�eorem 3.1.1. �ree dimensional lens spaces L(p, q) and L(p′, q′) are homeomorphic if andonly if p = p′ and q ≡ ±q′±1(mod p).

�e solution of the classi�cation problemup to homotopy equivalence is due toWhitehead [58]:

�eorem 3.1.2. �ree dimensional lens spaces L(p, q) and L(p′, q′) are homotopy equivalent ifand only if p = p′ and qq′ = ±n2(mod p) for some n ∈ N.

In the sections that follow, we present the disk diagram of a link presented in the second model

of L(p, q) and continue by presenting two types of diagrams for links in the solid torus: thepunctured disk diagram and the arrow diagram. Taking the (standard) inclusion T ↪ L(p, q)by the third model of L(p, q), a diagram of a link in the solid torus, equipped with additionalReidemeister moves, becomes a diagram of a link in L(p, q).

3.2 Disk diagrams

We overview the construction of the disk diagram as it is constructed in [41]. To get the disk

diagram of a link in L(p, q), we take the second model of L(p, q) and mark with N and S re-spectively the north pole N = (0, 0, 1) and south pole S = (0, 0,−1) of B3; we label the equatorialdisk lying in the xy-plane by D. Let L be a link in L(p, q) and by potentially making a smallisotopy on L, we assume that:

• N ∉ L, S ∉ L,

• L ∩ ∂B3 consist of a �nite set of points,

• L is not tangent to ∂B3,

• L ∩ ∂D = ∅.

Let p ∶ B3 ∖{N , S}→ D be the projection de�ned by p(w) = c(w)∩D, where c(w) is the circle(possibly line) through N , w, and S (Figure 3.4).

S

N

w

p(w) D

Figure 3.4:�e projection p ∶ B3 ∖ {N , S}→ D.

32 3.2. Disk diagrams

Making this projection to a knot diagram, we mark over- and undercrossings, i.e. consider a

double point P ∈ p(L) and take the preimage p−1(P) = {P1, P2} of this point. Suppose P1 iscloser to N than P2 is. �e image of a small arc a1 ∋ P1 on L represents an overpass and theimage of a2 ∋ P2 represents an underpass.

Since we would like to be able to reconstruct a link L ⊂ L(p, q) = B3/∼ from the diagram up toisotopy, it is essential that we have knowledge of what hemisphere the endpoints L∩∂B3 belongto. We label by +1,+2, . . . ,+n the endpoints in the upper hemisphere and by −1,−2, . . . ,−nthe points in the lower hemisphere, respecting the rule +i ∼ −i. An example is presented inFigure 3.5.

(a) a link L

+1+2

−1−2

(b) the diagram of L

Figure 3.5: A link in L(p, q) and its diagram.

3.2.1 Reidemeister moves of the disk diagram

�ere are seven Reidemeister moves in a disk diagram of L(p, q): three classical Reidemeistermoves (R-I, R-II, and R-III) and four additional Reidemeister moves (R-IV, R-V, R-VI, and

R-VII) acting across the boundary of D as viewed in Figure 3.6 [41].−1−2

+2+1 ←→

(a) R-IV

−1−2+2+1 ←→

−2−1+1+2

(b) R-V

−1

+2

−2+1 ←→

−1

+2+1−2

(c) R-VI

−1+1 ←→

+ j−i

(d) R-VII

Figure 3.6: Additional Reidemeister moves in the disk diagram of L(p, q).

In the case ofRP3, themove R-VII enables us to switch between positive and negative endpoints,so there is no need to include the labels. Also, the move R-VI does not appear, since it involves

the case p > 2. �e additional Reidemeister moves in the case of RP3 thus simplify to movesR-IV and R-V presented in Figure 3.7 [16].


←→

(a) R-IV

←→

(b) R-IV

Figure 3.7: Additional Reidemeister moves in the disk diagram of RP3.

3.3 Punctured disk diagrams of the solid torus

Imagine a knot K in the solid torus T = A× I, with A being an annulus (Figure 3.8(a)). A punc-tured disk diagram of a knot K is the regular projection of K on A, keeping the information ofover- and undercrossings (Figure 3.8(b)). We resolve the inconvenience of drawing the annulus

by making a dot in the region of R2 ⊃ A where the inner component of ∂A lies on and assumethe outer component of ∂A lies in the unbounded region of D (Figure 3.8(c)).

(a) (b) (c)

Figure 3.8: Construction of a punctured disk diagram of a link in the solid torus.

�e Reidemeister moves of a punctured disk diagram of a knot in the solid torus correspond to

the classical ones (R-I, R-II, and R-III), except that we cannot perform any move through the

puncture.

3.4 Punctured disk diagrams of L(p, q)

Taking the third model of L(p, q), we move a link L ∈ L(p, q) into the �rst component T1 ofT1∪hp ,q T2 by isotopy. Since L now lies entirely in T1, we project L on the annulus A of T1 = A× I,such a diagram corresponds to the punctured disk diagram of a link in T1.

3.4.1 Reidemeister moves of the punctured disk diagram of L(p, q)

We equip a punctured disk diagram with an additional Reidemeister move SL also known as

the slidemove2.�is move arises from gluing the solid torus T2 to T1 via the boundary homeo-2We use SL instead of R-IV, since we will o�en use this move in the chapters that follow and would like to, for

the ease of reading, make it distinct from the other Reidemeister moves.

34 3.5. Arrow diagrams

morphism hp,q.�emove is presented in Figure 3.9 [24]. One can visualize this move by sliding

an arc of the link over the meridional disk of the solid torus T2 glued to T1.

SL←ÐÐ→

}p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭q

Figure 3.9:�e slide move for L(p, q).

3.5 Arrow diagrams

An arrow diagram [39] is obtained from a link L ⊂ T by cutting T along a meridional disk D,projecting the resulting vertical cylinderD×I onto the diskD×{0}while keeping information ofover- and undercrossings of the projection (Figure 3.10(a)).�e intersection points L∩(D×{0})project onto arrows in the diagram, pointing to the part of L that is close to D × {0} in thecylinder and away from the part of L that is close to D × {1} (Figure 3.10(b)). We assume, bymaking a small isotopy on L, that there are no singularities near the arrows. If we do not drawthe bounding disk D in the diagram, but place the diagram on the plane, we assume ∂D lies inthe unbounded region of the plane (Figure 3.10(c)).

By convention we denote a set of consecutive arrows lying on an arc by a single arrow with an

integer above it representing the number of arrows – a negative integer indicating the directions

of the arrows are reversed (Figure 3.11).


D

(a) (b) (c)

Figure 3.10: Construction of an arrow diagram.

∼3

∼−3

Figure 3.11: Joining arrows in a diagram.

3.5.1 Reidemeister moves of the arrow diagram of the torus

Along with the three classical Reidemeister moves, there are two additional moves that act

across the disk D of the projection. �ese two Reidemeister moves, "cancellation of arrows"R-IV and "pushing an arrow over an arc" R-V are presented in Figure 3.12.�e geometric inter-

pretations of these moves are evident from Figure 3.13 [39].

←→ ←→

(a) R-IV

←→

(b) R-V

Figure 3.12: Additional Reidemeister moves in the arrow diagram.

←→ ←→

(a) interpretation of R-IV

←→

(b) interpretation of R-V

Figure 3.13: Geometrical interpretations of moves R-IV and R-V in the arrow diagram.

36 3.6. Arrow diagrams of L(p, q)

3.6 Arrow diagrams of L(p, q)

As in the case of the punctured disk diagram, we can represent a link in L(p, q) with a diagramof a link in the solid torus. Such diagrams are again accompanied by the extra slide move SL

that arises from the boundary-gluing homeomorphism of the tori.

3.6.1 Reidemeister moves of the arrow diagram of L(p, q)

A (p, q)-curve inside D × S1 (Figure 3.14) can be viewed as a (q, 1)-curve in the disk D of anarrowdiagramwith an arrow on every 2qπ/p angle.�e result is a diagramwith ⌊p/q⌋+1 arrowson the outer arcs and ⌊p/q⌋ arrows on the inner arcs in such a way that there are p arrows intotal (i.e. (pmod q) outer arcs with ⌊p/q⌋ + 1 arrows and q − (pmod q) inner arcs with ⌊p/q⌋arrows), see Figure 3.15 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭p

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭q

Figure 3.14: A (p, q)-curve inside D × S1.

∼

⌊p/q⌋

⌊p/q⌋

⌊p/q⌋ + 1

⌊p/q⌋ + 1

Figure 3.15: A (p, q)-curve viewed as an arrow diagram.

�e SLmove in the punctured disk diagram of Figure 3.9 can be viewed in D×S1 as the move inFigure 3.16. Note that SL is constructed by attaching (or detaching) a (p, q)-curve to K.

Passing this construction to the arrow diagram, the slide move SL presents (de)attaching a

(p, q)-curve as presented in Figure 3.17.

For q = 1 the move SL specializes to the move presented in Figure 3.18.


K

SL←ÐÐ→

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

p

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

q

K

Figure 3.16:�e slide move in D × S1.

K′ SL←ÐÐ→ K′

⌊p/q⌋

⌊p/q⌋

⌊p/q⌋ + 1⌊p/q⌋ + 1

Figure 3.17:�e slide move in the arrow diagram of L(p, q).

We take a closer look on how the punctured disk diagrams and the arrow diagrams are re-

lated.

3.6.2 Transition from punctured disk diagrams to arrow diagrams

A punctured disk diagam of a link L can be viewed as a closed tangle in the punctured planesuch as the one presented in Figure 3.19(a). �e link L can therefore be presented as a closedtangle inside the solid torus as pictured in Figure 3.19(b). By rotating this tangle inside the

cylinder D × I used in the construction of the arrow diagram (Figure 3.19(c)), the tangle can beprojected to D with each strand, used to close the tangle, containing an arrow (since they passthrough the meridional disk D) as pictured in Figure 3.19(d). Applying R-V moves, all arrowscan be moved to the upper endpoints of the tangle (Figure 3.19(f)). Notice that we can isotope

the strands, so they go from the upper to the lower endpoints of T making a full negative twistin between (Figure 3.19(g)).

38 3.6. Arrow diagrams of L(p, q)

K′SL←ÐÐ→ K′

p

Figure 3.18:�e slide move in an arrow diagram of L(p, 1).

T

(a)

Ð→ T

(b)

Ð→ T

(c)

Ð→ T

(d)

Ð→

Ð→ T

(e)

Ð→ T

(f)

Ð→ T

(g)

Figure 3.19: Passing from the punctured disk diagram to an arrow diagram.

A similar construction can be used by making a full positive twist and close the tangle on the

right side.

3.6.3 Transition from arrow diagrams to punctured disk diagrams

An arrow diagram of a link L (Figure 3.20(a)) can be viewed as an almost �at diagram outsidesmall neighborhoods of the arrows, i.e. it lies in a thickenedD×{1} inD×S1.�e neighborhoodsof arrows correspond to vertical strands parallel to {∗} × S1, ∗ ∈ D. By rotating the diagramand closing the vertical strands in the solid torus (Figure 3.20(b)), we can project the L to a theannulus I × S1.�e result is presented in Figure 3.20(c).

(a)

Ð→

(b)

Ð→

(c)

Figure 3.20: Passing from the arrow diagram to the punctured disk diagram.

4The HOMFLYPT skein module of L(p, 1)

In this chapter we use arrow diagrams of links in L(p, 1) to compute the HOMFLYPT skeinmodule of L(p, 1): we show that it is a free R-module and we exhibit the sets of generatorsfor each p ≥ 2 [14]. We also conjecture about the HOMFLYPT skein module of L(p, q), withq > 1.

4.1 The HOMFLYPT skein module of the solid torus

We translate the free basis of the HOMFLYPT skein module S3(T) of the solid torus T (com-puted in [23, 55]) into diagrams with arrows and establish some properties of these diagrams in

the skein module (see Section 2.3.2 for the de�nition of the HOMFLYPT skein module).

�eorem 4.1.1. S3(T) is a free R-module, generated by B = {t i1k1 . . . tisks ∶ s ∈ N, k1 < ⋯ < ks ∈

Z ∖ {0}, i1, . . . , is ∈ N} ∪ {∅}, where ∅ is the empty knot. For k > 0, tk is the oriented knot inT representing k in π1(T) ≅ Z that has a diagram such as the one pictured in Figure 4.1(a). Fork < 0, tk is t∣k∣ with reversed orientation.

For example, t3 and t3t−1 are presented in Figures 4.1: in the punctured disk diagrams on thele� and in the arrow diagram on the right. Recall that the number above the arrow indicates

the number of consecutive arrows.

Let L be a link in the solid torus T . We denote by HT(L) the expression of L in S3(T). �usHT(L) is a linear expression in elements of B with coe�cients in the ring R.

In an arrow diagram, we will refer to a circle with possible arrows on it as an oval.

We conclude this section with some properties of S3(T) that will be needed later.

Denote by tn, n > 0, an oval with clockwise orientation and n clockwise arrows. For n < 0, lettn be t−n with reversed orientation.

Lemma 4.1.1. In S3(T) we can revert clockwise arrows on an oval in the sense that for n > 0

tn =∑iAiTi or t−n =∑

iA′iT ′

i ,

39

40 4.1. �e HOMFLYPT skein module of the solid torus

∼

3

(a) t3

∼

3

(b) t3 t−1

Figure 4.1: Two generators of S3(T).

for some Ai ,A′i ∈ R. Any Ti has the form t i1k1 . . . tisks , where k1 > 0, . . . , ks > 0 and i1k1+⋯+ isks = n.

Similarly, any T ′i has the form t i1k1 . . . t

isks , where k1 < 0, . . . , ks < 0 and i1k1 +⋯ + isks = −n.

Proof. �e proof is by induction on n. For n = 1, t1 is transformed into t1 with a sequence ofR-I, R-V, and R-I moves. Similarly, t−1 is transformed into t−1. Suppose that the statement of thelemma is true for k < n. Let c be an oval with n clockwise arrows on it. We can perform a R-Imove on c, then push one arrow on the kink by R-V and then use the HOMFLYPT relation:

nR-IÐ→

nR-VÐ→

n − 1

= v2n − 1

+ vz−1n − 1

.

We repeat this process until all arrows are pushed to the right side of the kink, which leaves us

with terms with either counter-clockwise arrows or at most n − 1 clockwise arrows. We applyinduction on the latter.

Lemma 4.1.2. Let D be a diagram of a link L with an oval c containing n arrows, n ∈ Z, and astrand s adjacent to c, that may contain arrows outside the drawn region (Figure 4.2). �e oval ccan be pushed over s in the sense that either

n

=n∑i=0

Aiin − i

orn

=n∑i=0

Aiin − i

holds in S3(T) for some Ai ∈ R, depending on the orientation of the components c and s.

Chapter 4. �e HOMFLYPT skein module of L(p, 1) 41

sn

c

Figure 4.2: A strand adjacent to an oval.

Proof. We consider the case where c is oriented counter-clockwise and s is oriented "upwards".For n = 0 the proof is trivial. For n > 0 we �rst push an arc of c over s by R-II and successivelyapply Reidemesiter moves and HOMFLYPT relations until there are no arrows le� on the le�

side of s:

n

R-VÐ→

n − 1

= v2n − 1

+ vzn − 1

,

where we resolve the last term by successively applying the following Reidemeister moves and

HOMFLYPT relations:

k l

= v2k l

+ vz

klR-VÐ→ v2

k − 1 l + 1

+ vz

kl

.

It follows that the link can be expressed as

n

=n∑i=0

Aiin − i

.

A similar calculation shows that the above formula holds for any n < 0.�e formula also holdsif we reverse both orientations of c and s. If c is oriented clockwise and s is oriented "upwards"or if c is oriented counter-clockwise and s is oriented "downwards", then one gets similarly forany n ∈ Z:

n

=n∑i=0

Ai i

n − i

.

Lemma 4.1.3 (1- and 2-gon lemma). Let D be a diagram of a link in T and suppose that either astrand of D forms a 1-gon or two strands of D form a 2-gon (Figure 4.3).�en D can be expressedin S3(T) with diagrams which have less crossings than D and no more arrows than D.

Proof. Let d be the disk bounded by a 1-gon or a 2-gon. Wemay assume that there are no 1-gonsor 2-gons inside d (in the sense that the discs they bound would be included in d). Otherwise,we consider the most nested 1-gon or 2-gon instead (with no 1-gons or 2 gons inside).

42 4.2. �e construction of H and the main theorem

d

(a) 1-gon

d

(b) 2-gon

Figure 4.3: n-gons.

Case 1. Suppose d bounds a 1-gon. By assumption, there may be only ovals inside d, which wepush outside of d using Lemma 4.1.2. We continue by pushing the arrows lying on the strandof the 1-gon outside of d using HOMFLYPT relations and R-V. Finally, we remove the crossingof the 1-gon with R-I.

Case 2. Suppose d bounds a 2-gon. If there are ovals inside d, we push them outside of d usingLemma 4.1.2. We continue by pushing the arrows lying on the strands of the 2-gon outside of

d using HOMFLYPT relations and R-V. If the interior of d is not empty, there must be a braidgoing from one strand of the 2-gon to the other. In this case we push the arrows on the braid

outside of d and continue by pushing the crossings of the braid outside of d using HOMFLYPTrelations and R-III. We are le� with a trivial braid whose strands can be pushed outside of dthrough the double points of the 2-gon using HOMFLYPT relations and R-III. Once there are

no strands inside of d, we can perform an R-II move (using possibly a HOMFLYPT relationbefore) and remove the two crossings of the 2-gon.

4.2 The construction of H and the main theorem

Let Bp = {t i1k1 . . . tisks ∶ s ∈ N, k1 < ⋯ < ks ∈ Z ∖ {0},− p

2< k1 < ⋯ < ks ≤ p

2, i1, . . . , is ∈ N} ∪ {∅}

and let D be the set of arrow diagrams of links in L(p, 1).

We will construct a function H ∶ D → RBp. It will be shown that H induces an isomorphismfrom S3(L(p, 1)) to RBp.

�e function H is de�ned on standard diagrams of elements of B, such as the one in Figure 4.1,and extended linearly to all diagrams of links in T by the formula H(D) = H(HT(D)), whereHT(D) is to be understood as a linear expression in standard diagrams of elements of B (andnot just a linear expression in these elements).

Let g be a standard diagram of t i1k1 . . . tisks ∈ B. Let m be the maximum of ∣ki ∣. If m < p/2, or if

m = p/2 and no ki equals −m, we de�ne H(g) = g. Otherwise, let c be a component t−m of gor, if there are no such components, let c be a component tm of g. Let g′ be obtained from g byperforming an SL move on the component c.

We say, that an arrow is bad if its orientation is opposite to the orientation of the componenton which it lies and good otherwise. Notice, that there are less arrows in g′ than in g or, ifthis number is unchanged, there are less bad arrows in g′ than in g (this can occur only whenm = p/2). We recursively de�ne H(g) = H(HT(g′)). From Lemma 4.1.1 and Lemma 4.1.2 itfollows that no diagram of HT(g′) has more arrows than g′. Note that HT(g′) is computed by


pushing ovals out of the oval c transformed by SL, then possibly changing clockwise arrows intocounter-clockwise arrows. Following the proof of the lemmas, it is also clear that no additional

bad or good arrows appear in the computation of HT(g′). �us, all terms in HT(g′) have lessarrows than g or the same number of arrows but less bad arrows, and the recursive process inthe de�nition of H must end in �nitely many steps.

�eorem 4.2.1. H induces an isomorphism from S3(L(p, 1)) to RBp. �us, S3(L(p, 1)) is a freeR-module with the basis Bp.

Proof. HT is invariant under all Reidemeister moves except SL, so H is also invariant underthese moves. Also, HT satis�es the HOMFLYPT relation, so H satis�es it as well. H is invariantunder some SLmoves, namely the ones between pairs of diagrams g and g′ used in the de�nitionof H. �us, H will be an isomorphism from S3(L(p, 1)) to RBp provided it is invariant under

all SL moves.�is fact will be proven in the next section.

4.3 Invariance of H under SL moves

Let D be a diagram consisting of an element of B to which an oval e is added in such a way thatthere are no nested ovals in D.�e oval e has b ∈ Z arrows and it may have any orientation. LetD′ be obtained from D by an SL move performed on oval e. We call such a move from D to D′

standard (see Figure 4.4).

k1 k2 ks be

(a) D

SLÐ→

p

k1 k2 ks b

(b) D′

Figure 4.4: A standard SL move.

Proposition4.3.1. H is invariant under all SLmoves if it is invariant under all standard SLmoves.

Proof. Let D be a diagram on which we mark by s a small part on one of its arcs where SL canbe performed on; s is chosen in such a manner that no arrows lie on it. Let D′ be obtained from

D by performing a SL on s. It is enough to prove that by leaving the small arc s stationary, wecan express D as a linear expression of elements of B with an added oval e on which s lies.

If D has at least one crossing, then it must contain a 1-gon or a 2-gon. If s lies on it, then it iseasy to check, that there must be another 1-gon or 2-gon on which s does not lie (s also doesnot lie in the disc this 1-gon or 2-gon bounds, otherwise one could not perform an SL move on

it). One may therefore apply lemma 4.1.3 to this 1-gon or 2-gon not containing s. In this way wewill eventually remove all crossings in D. Using Lemma 4.1.2, we push all ovals out of the ovalon which s lies. Using Lemma 4.1.1, we revert the ovals with clockwise arrows on them, exceptfor the oval on which s lies. All these operations (removal of crossings, pushing and reverting

44 4.3. Invariance of H under SL moves

of ovals) can be performed simultaneously on D and D′. In this way the SL move from D to D′

can be expressed with several standard SL moves.

Proposition 4.3.2. H is invariant under all standard SLmoves.

Proof. We denote by SLn a standard SL move from D to D′ (such as in Figure 4.4), where one

of the diagrams has n arrows and the other one has at most n arrows. Here, we are counting allthe arrows as positive, even clockwise arrows (which may only lie on the component on which

the move is performed).

Let SLn be a standard move from D to D′ and let e be the component of D on which the moveis performed. �e proof that H(D) = H(D′) is by a double induction, on n and l , where l isthe number of arrows on the components of D except for e.

For n = 0 there are no SLmoves, so the hypothesis is true. Suppose that n > 0 and the hypothesisholds for all k < n. Notice, that if l = 0 there are no arrows on the components outside of e,so these components are mapped by HT to elements in the ring R, namely (v−1 − v)z−1, andH(D) = H(D′) holds by de�nition.

�ere are 3 cases to consider.

Case 1. D has n arrows and D′ has less than n. If e is a component of D with the maximalnumber of arrows and, moreover, bad arrows if there is any other component of D the maximalnumber of bad arrows on it, then H(D′) = H(D) by de�nition (see Figure 4.5(a) and 4.5(b),where only the ovals in question are pictured).

Otherwise, there is a component d of D such that H(D) = H(D′′) by de�nition, where D′′ is

obtained from D by performing an SL move on d (see Figure 4.5(c)). In this case, either d hasmore arrows than e, or it has the same number of arrows as e, but the arrows on d are bad andthe arrows on e are good.

a

d

b

e

(a) D

a p − b

(b) D′

b p − a

(c) D′′

Figure 4.5: Diagrams of D, D′, and D′′.

Let a and b be the numbers of arrows on d and e, respectively. Let E′ be the diagram obtainedfrom D′ by applying R-II and SL moves on d (see le� side of Figure 4.6). Similarly, let E′′ be thediagram obtained from D′′ by applying R-II and SL moves on e (see right side of Figure 4.6).Now, E′ and E′′ are diagrams of the same link in the torus T : one gets E′′ from E′ by applying2p times an R-V move on the right crossing of E′, thus pushing p arrows next to the a arrowsand p arrows away from the b arrows. It holds that H(E′) = H(E′′).

To show that H(D′) = H(E′), we �rst transform D′ and E′ with R-II and R-V moves into D′t

and E′t as shown in Figure 4.7.


a

p

p − b

(a) E′

b

p

p − a

(b) E′′

Figure 4.6: Diagrams of E and E′′.

a

p − b

(a) D′

a − ⌊ p+12⌋

⌊ p−12⌋ ⌊ p

2⌋ + 1 − b

⌊ p+12⌋

(b) D′t

a

p

p − b

(c) E′

a − ⌊ p+12⌋

⌊ p2⌋ + 1 − b

⌊ p2⌋⌊ p−1

2⌋

(d) E′t

Figure 4.7: Diagrams of D′t and E′t .

As D′′ and D′ have less arrows than D, we have a ≥ b > p/2. On D′t the number of arrows on

the component e is ⌊ p−12⌋ + ∣⌊ p

2⌋ + 1 − b∣ = ⌊ p−1

2⌋ + b − 1 − ⌊ p

2⌋, which equals b − 1 if p is odd and

b−2 if p is even. On component d there are a arrows.�us the number of arrows on D′t is n− 1

or n − 2. Similarly, on E′t there are less than n arrows: b − 1 or b − 2 on e and a or a − 1 on d.

We claim that H(E′t) = H(D′t). Indeed, using the same methods as in Lemma 4.1.2, one can

push out the a−⌊ p+12⌋ arrows on d, push any ovals out of e and, using Lemma4.1.1, revert all ovals

with clockwise arrows except for d, all this being donewithout increasing the number of arrows.�us, D′

t and E′t are simultaneously expressed with diagrams with no crossings, D′t = ∑i AiD′

iand E′t = ∑i AiE′i , where Ai ∈ R and E′i is obtained from D′

i with a standard SL move. As the

number of arrows in all D′i and E′i is less than n, we have H(D′

i) = H(E′i) by induction on n.�us H(E′) = H(E′t) = H(D′

t) = H(D′).

46 4.3. Invariance of H under SL moves

A similar argument shows that H(E′′) = H(D′′). Indeed, by exchanging the roles of a and band switching both crossings (which can be done by going around a component with an arrow

and applying R-V twice), E′′ is obtained from E′.

Combining preceding equalities, one gets H(D′) = H(D′′) = H(D).

Case 2. D has less than n arrows and D′ has n arrows. Recall that l is the number of arrows inD outside e. If l > 0, there are ovals with arrows other than e. We push these ovals through eat the same time in D and D′ (i.e. in D we push them into e and in D′ we push them out of e).From Lemma 4.1.2, in the resulting linear expressions for D and D′ two situations can occur:

all ovals are pushed through with their arrows, and we can apply Case 1 by reverting the roles of

D and D′; some arrows are transferred onto e, which decreases l and we apply the induction onl . In the second case it may happen that the total number of arrows is decreased and the globalinduction on n can be applied.

a

p/2

(a) D′

a − p/2 − 1

p/2

p/2 + 1

(b) D′t

a

p

p/2

(c) E′

a − p/2 − 1

p/2 − 1p/2

(d) E′t

Figure 4.8: Diagrams of D′, D′t , E′, and E′t .

Case 3. Both D and D′ have n arrows (this occurs when p is even and the number of arrows one is b = p/2). Two subcases are possible:

Case 3.1. �ere is a component d in D with a arrows, a > b = p/2, such that H(D) = H(D′′)by de�nition, where D′′ is obtained from D by performing an SL move on d. Using the samenotations as in Case 1, one gets again H(E′) = H(E′′) (see Figure 4.6).

Showing that H(D′′) = H(E′′) is done exactly as in Case 1: in both cases in order to applyinduction on n we only use the fact that a > p/2 (b can be equal to p/2).

Using several R-Vmoves D′ and E′ can be transformed simultaneously into diagrams D′t and E′t


shown in Figure 4.8. Notice that the number of arrows in D′t is n (because a > p/2 by assump-

tion) and the number of arrows in E′t is less than n. As in Case 1, D′t and E′t are simultaneously

expressed with diagrams with no crossings, D′t = ∑i AiD′

i and E′t = ∑i AiE′i , where Ai ∈ R andE′i is obtained from D′

i with a standard SLmove. As the number of arrows in all E′i is less than n,we have H(D′

i) = H(E′i) by Case 1, if there are n arrows in D′i , or by induction on n otherwise.

�us H(D′t) = H(E′t) and H(D′) = H(D′

t) = H(E′t) = H(E′).From preceding equalities it follows that H(D′) = H(D′′) = H(D).Case 3.2. �ere are no components in D with more than p/2 arrows. If the number of badarrows is decreased from D to D′, then H(D) = H(D′) by de�nition. Otherwise, we use againinduction on l . If l > 0, we push all the ovals into e in D (and out of e in D′), as it was done

in Case 2, obtaining again two situations: all ovals with their arrows are pushed and H(D′) =H(D) by de�nition; some arrows are transferred onto e. In the last case, if the number ofarrows on e is still p/2, induction on l is applied and if this number is changed, we get Case 1or Case 2.

4.4 The case of L(p, q)

For the case of L(p, q), we construct a function H ∶ D → RBp,q, where

Bp,q = {t i1k1 . . . tisks ∶ s ∈ N, k1 < ⋯ < ks ∈ Z ∖ {0},−p/2 < k1 < ⋯ < ks ≤ p/2, i1..is ∈ N} ∪ {∅},

andH is de�ned analogously to that of the case of L(p, 1). In short, ifHT is the linear expression

in standard diagrams of elements of the basis B of S3(T) and if g is a diagram of t i1k1 . . . tisks ∈ B

and m =max{∣ki ∣}i , then H is de�ned recursively as:

H(g) =⎧⎪⎪⎨⎪⎪⎩

g , g ∈ Bp,q

H(HT(g′)), otherwise,

where g′ is the diagram obtained by performing a SL move on the component t−m of g, or, if nosuch components exist, SL is performed on the component tm.

With similar arguments than in the L(p, 1) case, it can be shown that the recursive process endsin �nitely many steps, so the only thing le� to show is that H is invariant under SL moves.Unfortunately the SL moves produce diagrams that are too complicated to apply the methods

described in the previous section.

Conjecture 4.4.1. �e module S3(L(p, q)), q > 1, is a free R-module with the basis:

Bp,q = {t i1k1 . . . tisks ∶ s ∈ N, k1 < ⋯ < ks ∈ Z ∖ {0},−p/2 < k1 < ⋯ < ks ≤ p/2, i1, . . . , is ∈ N} ∪ {∅}.

5The categorification of the Kauffman

bracket skein module of RP3

Various generalizations of the Khovanov homology have been constructed so far. One gener-

alization was done by Asaeda, Przytycki, and Sikora [3], who generalized the construction by

de�ning a double graded homology theory that categori�es the Kau�man bracket skein mod-

ule of links in I-bundles over surfaces. �eir construction did not work for the twisted bun-dle RP2×I, the problem being in essence the strange behavior of links projected to the non-orientable surface RP2. Another generalization was done by Manturov, who managed to cate-gorify the Jones polynomial of virtual links [37], which, as a special case, includes the categori-

�cation of the Jones polynomial of links in RP3.

�is section provides details and explicitly shows how to categorifyKBSMof links inRP3, whichis equivalent to categorifying KBSM of the twisted I-bundle overRP2 (RP2×I ≈ RP3∖{∗}), themissing piece of the puzzle in [3]. At the same time, this categori�cation will also provide a

strong invariant of links in RP3.

Disk diagrams are used throughout this section.

5.1 The Kauffman bracket skein module of RP3

It is shown in [24] that the Kau�man bracket skein module of the lens space L(p, q) is a freeR-module with ⌊p/2⌋ + 1 generators, in particular, S2,∞(RP3) has two generators, the isotopyclass of the empty set [∅] and the class of the orientation reversing curve [x] of RP2 ⊂ RP3,shown in Figure 5.1 [24]1.

Using notation as in Section 2.3.1, the state-sum formula for expressing [L] ∈ S2,∞(RP3) in theabove basis is given by the formula

S2,∞(RP3)([L]) = ∑s∈{0,1}X

A#0(s)−#1(s)(−A2 − A−2)∣s∣T [x]∣s∣P[∅]1−∣s∣P ,

1In [24] they use an equivalent basis {[O], [x]}.

49

50 5.2. �e chain complex

x

Figure 5.1: A generator of KBSM of RP3.

where ∣s∣T denotes the number of a�ne trivial circles in s and ∣s∣P denotes the number of pro-jective circles (i.e. non-a�ne trivial circles, such as the one in Figure 5.1) in s. In the aboveformula we o�en use the evaluation [∅] = 1 and replace [x] by a variable x.

5.2 The chain complex

As we have alreadymentioned, Asaeda et al. managed to construct a chain complex of bigraded

vector spaces that categorify KBSM of I-bundles over surfaces [3], where their constructionfails for RP2×I due to strange behavior of links projected to RP2 (i.e. the existence of 1 → 1bifurcations which will be described latter). Manturov managed to overcome the problem of

1→ 1 bifurcations when he categori�ed the Jones polynomial of virtual links [37].In this section we de�ne the Khovanov chain complex with gradings similar to those de�ned

by Asaeda et al., but in the next section, we will rede�ne their di�erential using techniques of

Manturov to control the odd behavior of circles mentioned above.

Since, in this chapter, we will construct the chain complex as a chain complex of Z⊕Z-gradedZ-modules, we adjust most of the vocabulary made in section 2.4 accordingly.

LetW =⊕ j,k Wj,k be a Z⊕Z-graded Z-module.�e Poincaré polynomial of a Z⊕Z-gradedZ-module in variables A and z is de�ned as

P(W) =∑j,k

Ajzk rankWj,k .

�e standard construction of the Khovanov homology, as well as the construction in [3], uses

tensor products to form chain complexes, we, however, use thewedge product (which is in some

sense an ordered version of the tensor product).

�e wedge product W ∧ W ′ of the modules W = ⊕ j,k Wj,k and W ′ = ⊕ j,k W ′j,k is de�ned

as

W ∧W ′ = ⊕j= j1+ j2k=k1+k2

Wj1 ,k1 ∧W ′j2 ,k2 .

�e wedge product is associative and anticommutative.

For a permutation σ ∈ Sn and for elements w1 ∈ W1,w2 ∈ W2, . . . ,wn ∈ Wn, the wedge product

is subject to the permutation rule:

wσ(1) ∧wσ(1) ∧⋯ ∧wσ(n) = sign(σ)w1 ∧w2 ∧⋯ ∧wn .

Chapter 5. �e categori�cation of the Kau�man bracket skein module of RP3 51

�e degree shi� operator q{l ,m} shi�s the gradings ofW =⊕ j,k Wj,k by (l ,m):

W{l ,m} =⊕j,k

Wj−l ,k−m .

�e degree shi� q{l , 0} is abbreviated to q{l}, such a shi� corresponds to multiplication by Al

in the Poincaré polynomial: P(W{l}) = AlP(W).Let V = ⟨1, X⟩2 be the bigradedZ-module freely generated by elements 1 and X with bigradingsdeg 1 = (−2, 0) and degX = (2, 0), and let V = ⟨1, X⟩ be the module generated by 1 and X withdeg 1 = (0, 1) and degX = (0,−1). Note that P(V) = A2 + A−2 and P(V) = z + z−1, which isexactly what we assign a circle (trivial or projective, respectively) in the state-sum formula of

KBSM.

Recall fromSection 2.2 that for the set of crossingsX = {1, 2, . . . , n} of the diagramD of a link L,a state s is an element of the discrete cube {0, 1}X and corresponds to a complete smoothening ofD.�e number of 0 and 1 factors in s is denoted by #0(s) and #1(s), respectively. Assuming thecircles are enumerated in a way that the possible projective circle is at the end, Cs will represent

the module associated with the state s. A circle in s, enumerated by i, contributes a factor Viif the circle is trivial or a factor V i if the circle is projective. �e module Cs is therefore either

equal to

Cs = (V1 ∧ V2 ∧⋯ ∧ Vn){#0(s) − #1(s)}or is equal to

Cs = (V1 ∧ V2 ∧⋯ ∧ Vn−1 ∧ V n){#0(s) − #1(s)},depending on whether s has a projective circle or not.

�e module of i-chains is now de�ned to be the sum of all Cs’s where the di�erence between

the number of 0 and 1 factors in s is i:

Ci = ⊕s∈{0,1}X

#0(s)−#1(s)=i

Cs .

Forming such direct sums of modules is sometimes called a �attening of {Cs}s . We claimthat

0Ð→ Cn Ð→ Cn−2 Ð→ ⋯Ð→ C−n+2 Ð→ C−n Ð→ 0with suitable di�erentials yet to be de�ned, forms a Khovanov chain complex.

5.3 The differential

Let ξ ∈ {0, 1, ⋆}X be a sequence of length n with the property that ⋆ appears only once in ξ. Byreplacing ⋆ by 0 we get a vertex of the discrete cube {0, 1}X denoted by ξ⋆→0 and by replacing ⋆by 1 we get an adjacent vertex ξ⋆→1. We call such a ξ an edge of {0, 1}X . Recall that each vertexcorresponds to a state of the diagram, that being so, each edge corresponds to a local "change"

2�e symbols 1 and X play a similar role than v− and v+ in Section 2.4.�e choice of using 1 and X instead ofv± is just a personal preference of the author and is the notation used in several papers.

52 5.3. �e di�erential

of two adjacent states. Each ξ is associated with a linear map dξ ∶ Cξ⋆→0 → Cξ⋆→1 called a partialdi�erential, which will be de�ned in the next few paragraphs.

For a diagram D, we call the collection of modules {Cs}s∈{0,1}X , together with their partial dif-ferentials, the cube JDK of D. Now, JDK needs to form an anticommutative diagram, since thisforces the �attened JDK to form a well-de�ned chain complex (C●, ∂●). As seen in [3], there isno obvious way to de�ne the partial di�erential for links in RP3, but as shown in [37] this ispossible to achieve for the Jones polynomial, if the links and the circles in the states are oriented

and certain signs are applied to the partial di�erentials.

Hence, we orient both the diagram D and the circles of the states in an arbitrary manner. In theneighborhood of a crossing there are either one or two circles. If we change the smoothening

in a crossing from type 0 to type 1, we call the change of the involved circles a bifurcation. Itis evident that the circles involved in the states ξ⋆→0 and ξ⋆→1 are in a bijective correspondence,with one of these exceptions (Figure 5.2):

1. two circles in ξ⋆→0 join into one circle in ξ⋆→1 (type 2→ 1 bifurcation),

2. a circle in ξ⋆→0 splits into two circles in ξ⋆→1 (type 1→ 2 bifurcation),

3. a circle in ξ⋆→0 twists into a circle in ξ⋆→1 (type 1→ 1 bifurcation).

Ð→

(a) type 2→ 1

Ð→

(b) type 1→ 2

Ð→

(c) type 1→ 1

Figure 5.2:�ree types of bifurcations.

Note that the type 1 → 1 bifurcation can only appear when the circle lies within a Möbius bandand can therefore only appear in the case of projecting to a non-orientable surface, such asRP2(see Figure 5.3).

(a) L (b) state 0 of L (c) state 1 of L

Figure 5.3:�e appearance of a 1→ 1 bifurcation on a Möbius band.

Looking at a crossing c of D in such a way that the two outgoing arcs are facing northwestand northeast and that the ingoing arcs are facing southwest and southeast, we call a circle of

some state locally consistently oriented at c if its orientation agrees with the orientation ofthe northeast arc or disagrees with the orientation of the southwest arc (with respect to the

arc before the smoothening); vice versa, a circle is locally inconsistently oriented at c if itsorientation disagrees with the northeast arc or agrees with the southwest arc (see Figure 5.4). It


can happen that both arcs in the state belong to the same circle and the consistency cannot be

uniquely de�ned, in this case the local consistency is indetermined at c.

Figure 5.4: Locally consistent orientations.

In order to de�ne the di�erential, we rearrange the∧-factors ofCξ⋆→0 by a permutation σ ∈ S∣ξ⋆→0 ∣in such a way that the factors involved in the bifurcation are at the beginning of the wedge

product. Furthermore, if the bifurcation is of type 2→ 1, assuming that the arcs of the diagramare facing northwest and northeast, we wish that the �rst factor in the domain is represented by

a "le�" circle if the bifurcation site is at a positive crossing and is represented by a "top" circle if

the bifurcation site is at a negative crossing (see Figure 5.5).

1 2mÐ→

∆Ð→2

1

(a) positive crossing

2

1mÐ→

∆Ð→ 1 2

(b) negative crossing

Figure 5.5: Order of (locally consistently oriented) circles of a bifurcation.

If the bifurcation is of type 2 → 1, wemultiply the �rst two factors; if the bifurcation is of type1 → 2, we comultiply the �rst factor; and if the bifurcation is of type 1 → 1 we apply the 0 mapto the �rst factor. On the remaining factors we apply the semi-identity I = I ∧ I ∧ ⋯ ∧ I (∧ I),where the map I ∶ V → V (resp. I ∶ V → V ) is de�ned by I(1) = 1, I(X) = ±X (resp. I(1) =1, I(X) = ±X), with a plus sign on the generator X if the circles that twoV ’s (resp. V ’s) representhave the same orientations and with a minus sign if the orientations are opposite.

Letm be themultiplicationoperator and ∆ the comultiplicationoperator, they are both linearmaps subject to the rules in Table 5.1.

As suggested in Figure 5.5, the order of the factors in the codomain of ∆ depend on the position

of the circles they represent. At a positive crossing the �rst factor is presented by the "top" circle

and at a negative crossing the �rst factor is represented by a "le�" circle.

�e sign of each factor X (resp. X) in Table 5.1 depends on the local consistency of the circlethat X (resp. X) represents at the crossing where the bifurcation appears. If the circle is locallyconsistently oriented at the crossing, the sign is positive and if the circle is locally inconsistently

oriented, the sign is negative. Note that the consistency is indetermined only at bifurcations of

type 1→ 1.

54 5.3. �e di�erential

Table 5.1: Multiplication and comultiplication (shi�s omitted).

m ∶ V1 ∧ V2 Ð→ V ∆ ∶ V Ð→ V1 ∧ V211 ∧ 12 z→ 1 1 z→ 11 ∧ (±X2) + (±X1) ∧ 1211 ∧ (±X2)z→ (±X) (±X)z→ (±X1) ∧ (±X2)

(±X1) ∧ 12 z→ (±X)(±X1) ∧ (±X2)z→ 0

m ∶ V 1 ∧ V2 Ð→ V ∆ ∶ V Ð→ V 1 ∧ V211 ∧ 12 z→ 1 1 z→ 11 ∧ (±X2)11 ∧ (±X2)z→ 0 (±X)z→ (±X1)∧ (±X2)

(±X1)∧ 12 z→ (±X)(±X1)∧ (±X2)z→ 0

m ∶ V1 ∧ V 2 Ð→ V ∆ ∶ V Ð→ V1 ∧ V 211 ∧ 12 z→ 1 1 z→ (±X1) ∧ 12

11 ∧ (±X2)z→ (±X) (±X)z→ (±X1) ∧ (±X2)(±X1) ∧ 12 z→ 0

(±X1) ∧ (±X2)z→ 0

A�er (co)multiplying and applying the identity, we rearrange the factors in the result by the

permutation ρ ∈ S∣ξ⋆→1 ∣, so that they agree with the order of factors in the codomain Cξ⋆→1 .

�e above steps describe the partial di�erential dξ ∶ Cξ⋆→0 → Cξ⋆→1 . In detail, if Pσ is a map

that rearranges factors in Cξ⋆→0 by σ and Pρ is the map that rearranges factors in Cξ⋆→1 by ρ,then

dξ =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Pρ ○ (m ∧ I) ○ Pσ , if ∣ξ⋆→0∣ = ∣ξ⋆→1∣ + 1,Pρ ○ (∆ ∧ I) ○ Pσ , if ∣ξ⋆→0∣ + 1 = ∣ξ⋆→1∣,Pρ ○ (0 ∧ I) ○ Pσ , if ∣ξ⋆→0∣ = ∣ξ⋆→1∣.

(5.1)

�e total di�erential ∂i ∶ Ci → Ci−2 is the sum of partial di�erentials:

∂i = ⊕ξ∈{0,1,⋆}X

#0(ξ)−#1(ξ)=i−1

dξ .

We shall call such a sum a �attening of partial di�erentials.

As an illustration of how the partial and total di�erentials work, a detailed example is presented

at the end of the next section.


5.4 The homology

�e i-th Khovanov homology group is

Hi =Ker ∂iIm ∂(i+2)

.

�eorem 5.4.1 (proof in section 5.5). Let L be a link inRP3 and D a diagram of L.�en ∂○∂ = 0so that (C●, ∂●) is a chain complex.

�eorem 5.4.2 (proof in section 5.5). For a diagram D of the link L ⊂ RP3,H●(D) is preservedunder R-II, R-III, R-IV, and R-V.�ereforeH● is an invariant of framed links in RP3.

�e Euler characteristic of a Z3-graded Z-moduleW =⊕i , j,k(Wi) j,k is for the purpose of thisthesis de�ned to be

χ(W) = ∑i , j,k

(−1)j−i2 Ajzk rank(Wi) j,k .

Sincewehave adjusted our gradings so that they agreewith the state-sum formula L inS2,∞(RP3),it follows from the construction that χ(C●(D)) = χ(H●(D)) = S2,∞(RP3)(L), where the lastequation holds by substituting z + z−1 with [x].

(a) link L

1

2

00

10

01

1

2

11

d⋆0

d0⋆

d1⋆

d⋆1

(b) cube of L

Figure 5.6:�e link L and its cube.

Example 5.4.3. In the link in Figure 5.6 we get:

C00 = V1 ∧ V 2{2}, C01 = V , C10 = V , C11 = V1 ∧ V 2{−2}.

Omitting shi�s, partial di�erentials work in the following manner:

d0⋆(11 ∧ 12) = 1, d0⋆(−11 ∧ X2) = X, d0⋆(X1 ∧ 12) = 0, d0⋆(−X1 ∧ X2) = 0;d⋆0(12 ∧ 11) = 1, d⋆0(−12 ∧ X1) = 0, d⋆0(X2 ∧ 11) = −X, d⋆0(−X2 ∧ X1) = 0;d⋆1(1) = X1 ∧ 12, d⋆1(−X) = −X1 ∧ X2; d1⋆(1) = −12 ∧ X1v, d1⋆(X) = −X2 ∧ X1.

56 5.5. Proofs

�e total di�erential is:

∂2(11 ∧ 12) = 1⊕ (−1), ∂2(11 ∧ X2) = (−X)⊕ X, ∂2(X1 ∧ 12) = 0⊕ 0,∂2(X1 ∧ X2) = 0⊕ 0; ∂0(1⊕ 0) = X1 ∧ 12, ∂0(X ⊕ 0) = X1 ∧ X2,∂0(0⊕ 1) = X1 ∧ 12, ∂0(0⊕ X) = X1 ∧ X2; ∂−2 = 0.

It clearly holds that ∂ ○ ∂ = 0.�e homology groups are:

H2 ≅ ⟨X1 ∧ 12, X1 ∧ X2⟩{2}, H0 ≅ 0, H−2 ≅ ⟨11 ∧ 12, 11 ∧ X2⟩{−2},

i.e. the only non-trivial dimensions ofH● are

(H2)4,1 ≅ (H2)4,−1 ≅ (H−2)−4,1 ≅ (H−2)−4,−1 ≅ Z.

�e Euler characteristic is

χ(H●) = (−A4 − A−4)(z + z−1) = (−A4 − A−4)[x].

5.5 Proofs

Lemmas 5.5.1 – 5.5.3 will show that the homology is independent of the free choices wemade on

choosing the orderings of circles, the orientations of the link, and the orientations of the circles

in the states, respectfully.

Lemma 5.5.1. �e homology is invariant under the ordering of circles.

Proof. For each state s of D, let os and o′s be two di�erent orderings of circles in s. Let os ando′s di�er by a permutation πs. Permutations πs induce isomorphisms Pπs ∶ Cs → Cs on the

associated modules. Let, for two adjacent states q = ξ⋆→0 and r = ξ⋆→1, the partial di�erentialdξ ∶ Cq → Cr be de�ned in terms of the �rst ordering as Pρr ○ d ○ Pσq , where in place of d’s wehave either m ∧ I , ∆ ∧ I , or 0. It follows from the de�nition that the di�erential in terms of thesecond ordering equals Pπrρr ○d ○Pσqπ−1q . Since Pπrρr ○d ○Pσqπ−1q ○Pπq = Pπr ○Pρr ○d ○Pσq , it follows

that the maps {Pπs}s form a chain map.

Lemma 5.5.2. �e homology is invariant under the change of link orientation.

Proof. It is enough to prove invariance under changing the orientation of one component. Letc be a crossing of the changed component. If c is a self-crossing, changing the orientationof both strands of c has the same e�ect as transposing the orderings of the circles that splitat c (see Figure 5.7), which is invariant by Lemma 5.5.1. If c is not a self-crossing, changingthe orientation of one component leads to inverting the locally consistent orientation on all

participating circles and transposing the ordering of the two circles if the bifurcation site is

of type 1 → 2 (see Figure 5.8); by writing down the partial di�erentials before and a�er theorientation change, one has to check that they both agree.

We check the di�erential at a 1 → 2 bifurcation in the case of a negative crossing, other casesare easier to check and are treated similarly.


�e partial di�erential before the change is the map:

∆ ∶ 1 z→ 11 ∧ (±X2) + (±X1) ∧ 12,±X z→ ±X1 ∧ ±X2,

a�er the change of orientation the map agrees with the above (see Figure 5.8):

∆ ∶ 1 z→ 12 ∧ (∓X1) + (∓X2) ∧ 11 = 11 ∧ (±X2) + (±X1) ∧ 12,±X z→ −(∓X2 ∧ ∓X1) = ∓X1 ∧ ∓X2 = ±X1 ∧ ±X2.

2

1mÐ→

∆Ð→ 1 2

(a) possible positions of circles at c

1

2mÐ→

∆Ð→ 2 1

(b) crossing c with changed orientation

Figure 5.7: Order of locally consistently oriented circles before and a�er a change of orientation

at a self-crossing.

2

1mÐ→

∆Ð→ 1 2

(a) possible positions of circles at c

2

1mÐ→

∆Ð→ 2 1

(b) crossing c with a changed orientation

Figure 5.8: Order of locally consistently oriented circles before an a�er a change of orientation

of a component at a crossing.

Lemma 5.5.3. �e homology is invariant under the change of orientations of the circles in thestates.

58 5.5. Proofs

Proof. Let fs ∶ Cs → Cs be the semi-identity that sends X to −X and 1 to 1 (resp. X to −X and 1to 1) on the V (resp. V ) components of Cs that represent circles with changed orientations. We

observe that changing the orientation of a circle changes the local consistency in the de�nition

of the partial di�erential d. But this is exactly the opposite of what f does, so f ○d ○ f −1 = d .

Proof of theorem 5.4.1. To show that ∂○∂ = 0 holds for a projective link diagramD, it is su�cientto prove that every 2-dimensional face of JDK is anticommutative. Every 2-dimensional facecorresponds to a smoothening of n−2 crossings, since the remaining two crossings are resolvedin four di�erent ways. Since the circles that are non-adjacent to either of the two crossings are

mapped by I , they can be omitted in the proofs. It therefore su�ces to prove anticommutativityfor all possible 2-crossing projective diagrams. For constructing all such diagrams, we take a

similar approach as in [3, 37]. Here is an outline.

Take all 4-valent graphs in RP2 with 2 rigid vertices and replace each vertex with either anovercrossing or an undercrossing. �is produces all possible 2-crossing diagrams. Certain

graphs can be omitted due to symmetries that leave circles in a natural bijective correspon-

dence preserved under partial di�erentials. Take vertices v and u and enumerate their edges byv0, v1, v2, v3, and u0, u1, u2, u3 (Figure 5.9).

v0 v1 u0 u1

v3 v2 u3 u2

Figure 5.9: Vertices v and u.

We disregard non-connected graphs, since it clearly holds that in this case either d0⋆ = −d1⋆,d⋆0 = d⋆1 or d0⋆ = d1⋆, d⋆0 = −d⋆1. Without loss of generality, wemay assume that v0 is connectedto v1. Two graphs are symmetric if they are related by the following operations [3]:

1. Exchanging v and u produces the same graph.

2. A cyclic permutation vi → vi−k mod 4 and ui → ui−k mod 4 for i ∈ {0, 1, 2, 3} if vk and uk are

connected for some k ∈ {0, 1, 2, 3}.3. A �ip: either vi ↔ vi+2 mod 4 or ui ↔ ui+2 mod 4 are exchanged for some i ∈ {0, 1, 2, 3}.

�e last relation follows from the fact that if link diagrams D1 and D2 are represented by graphsΓ1 and Γ2 related by a �ip, then there is a natural bijective correspondence between the states of

D1 and the states of D2.�is correspondence is preserved by changing of smoothing and partialdi�erentials.

Lemma 5.5.4. �ere are exactly 6 non-symmetric graphs in RP2. �e graphs are represented inFigure 5.10.

To list all the possible 2-crossing links, we place all 4 possible combinations of crossing on each

vertex of the above graphs. We can omit links where the 1 → 1 bifurcation appears in each ofthe composites of the partial di�erentials, since in this case, it clearly holds that ∂ ○ ∂ = 0.


(a) (b) (c)

(d) (e) (f)

Figure 5.10: Non-symmetric 4-valent connected graphs with 2 vertices in RP2.

Lemma 5.5.5. Up to symmetry and disregarding non-essential cases with two 1 → 1 bifurcationsappearing on opposite sides of the composites, there are 11 projective links with two crossings thatare represented in Figure 5.11.

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k)

Figure 5.11: Non-symmetric essential links with 2 crossings in RP2.

�e proofs of Lemma 5.5.4 and Lemma 5.5.5 rely on an extensive case-by-case checking.

60 5.5. Proofs

(a) link L

00

10

1 2

01

11

d⋆0

d0⋆

d1⋆

d⋆1

(b) cube of L

Figure 5.12:�e link L of Figure 5.11(a) and its cube.

Due to Lemma 5.5.1, Lemma 5.5.2 and Lemma 5.5.3, we can orient the above 11 links and their

states arbitrarily, order the circles arbitrarily and calculate the di�erentials to show that ∂○∂ = 0.We check anticommutativity only for the link in Figure 5.11(a), the rest can be checked similarly.

From Figure 5.12 we get

d⋆0(1) = 12 ∧ X1 − X2 ∧ 11,d⋆0(−X) = −X2 ∧ X1;

d1⋆(11 ∧ 12) = 1,d1⋆(−11 ∧ X2) = X ,d1⋆(−X1 ∧ 12) = X ,d1⋆(X1 ∧ X2) = 0;

d0⋆ = d⋆1 = 0,

hence d⋆0d1⋆ = d0⋆d⋆1 = 0.

Proof of theorem 5.4.2. �e proofs of invariance under moves R-II and R-III almost entirelycoincide with those of the classical case [4]. �e main tool used for showing invariance is the

following cancellation principle:

Lemma 5.5.6. Let C be a chain complex and let C′ ⊂ C be a sub-chain complex. �en, if C′ isacyclic it holds that H(C) = H(C/C′), on the other hand, if C/C′ is acyclic it holds that H(C) =H(C′).

Proof. Both equalities follow trivially from the long exact homology sequence

⋯Ð→ Hn(C′)Ð→ Hn(C)Ð→ Hn(C/C′)Ð→ Hn+1(C)Ð→ ⋯

associated with the short exact sequence 0Ð→ C′ Ð→ C Ð→ C/C′ Ð→ 0.


Invariance under R-II. �e cube ofq y

can be expressed in terms of subcubes as indicated

on the le� diagram below.

r z{−2}

r z

r z

r z{2}

∆ m

C ⊃ 0

r z

1

0

r z{2}

m

C′

Let C be the �attening of this cube. �e chain complex C contains a subcomplex C′ in whichthe subscript 1 of

q y1denotes the cube of submodules where the trivial "middle" circle is not

assigned the module V , but instead the free module ⟨1⟩. Since m is an isomorphism in C′, C′ isacyclic andH(C) ≅ H(C/C′) by Lemma 5.5.6.

r z{−2}

r z

X

r z

0

∆

d⋆0

τC/C′

�e map ∆ is an isomorphism in C/C′, so we can de�ne a map τ that is the composition τ =d⋆0∆−1. Let C′′ be the subcomplex of C/C′ consisting of all elements ξ ∈

q yand all elements

of the form β⊕ τβ ∈q y

X ⊕q y

. Taking C/C′ mod C′′, we killq y

and impose the relation

β⊕ 0 = 0⊕ τβ inq y

X ⊕q y

, but we have an arbitrary choice of γ ∈q y

. Hence (C/C′)/C′′is isomorphic to

q y.

Invariance under R-III. In order to prove invariance under the third Reidemeister move, weexpand both cubes

r zand

r zto the le� and right diagrams below, respectfully.

r z

r z

r z

r z

r z

r z

r z

r z

∆

mr z

r z

r z

r z

r z

r z

r z

r zm∆

62 5.5. Proofs

In these diagrams we omit degree shi�s. For simplicity, we also keep the states of both cubes

coherently oriented.

We repeat the de�nitions of modding out acyclic complexes C′ and C′′ from the proof of R-IIwith the top-right parallelograms, as the result we obtain:

r z

0

r z

r z

r z

X

r z

r z

0

d1,⋆01

d1,⋆10

τ1r z

0

r z

r z

r z

r z

X

r z

0

d2,⋆01

d2,⋆10

τ2

Let T be the homomorphism that sends the bottom-le� parallelogram of the le� cube to thebottom-le� parallelogram of the right cube, but transposes the top-right parallelogram by send-

ing

r z

X⊕

r zto

r z

X⊕

r zvia T(β1⊕γ1) = β2⊕γ2 for β1 ∈

r z

X, γ1 ∈

r z, β2 ∈

r z

Xand γ2 ∈

r z. Carefully writing down the de�nitions of the partial di�erentials, it

easily follows that τ1 ○ d1,⋆01 = d2,⋆01 and d1,⋆10 = τ2 ○ d2,⋆10, hence, T is an isomorphism.Invariance under R-IV and R-V. Invariance under the Reidemeister moves R-IV and R-V fol-lows trivially, since there is a natural bijective correspondence that preserves the di�erentials

between states of the diagram before and a�er these moves.

6Classification

In this chapter we describe the computer algorithm used to classify non-a�ne prime knots in

the solid torus T and in the lens spaces L(p, q). We provide a full classi�cation of knots up tofour crossings in the above spaces. We also provide a classi�cation up to �ve crossings with a

few exceptions. We establish which of the knots are amphichiral in T . A knot in T or L(p, q)is considered prime if it cannot be written as a connected sum of two non-trivial knots, with

one of the knots in the composition being a�ne. �e results are presented at the end of the

chapter with details available in the Appendices. �e computer algorithm is available online

at [10].

6.1 Knot notation

Knot notations have been developed to describe knots in a way that the knot can be recon-

structed from the notation up to isotopy. Out of various notations, the Gauss code is perhaps

the most uninvolved one and is therefore the most suitable notation for storing and processing

knots by computer.

AGauss word of length n is a word on the alphabet {±1,±2, . . . ± n}, where each letter appearsexactly once.

LetD be a diagramof a knot in S3 with n crossings.�eGauss word associatedwith the diagramD is the Gauss word of length n obtained by the following steps [20, 49]:

1. Enumerate the crossings from 1 to n.

2. Orient the diagram (if not already oriented) and choose an initial point on it.

3. Starting with an empty word, travel from the initial point back to it according to the ori-

entation, appending the letter ±k to the end of the word when passing through a crossingk with a + sign if it is an overcrossing and a − sign if it is an undercrossing.

We use the convention that we place the initial point right before an overcrossing and num-

ber the crossings in such a way that the �rst encountered crossing is +1 and for each crossingencountered for the �rst time, the next available integer is used.

63

64 6.1. Knot notation

In general, we can reconstruct a knot from its associated Gauss word only up to its mirror

image, but having the knowledge of the knot’s crossings signs enables us to reconstruct the knot

completely.

A Gauss code is a Gauss word followed by a sequence of n signs.

�e Gauss code of a diagram D is the Gauss word of D followed by the sequence of crossingssigns (in the order of the enumeration).

Example 6.1.1. �e Gauss code of a diagram of the �gure eight knot pictured in Figure 6.1 is equalto

1 -2 3 -4 2 -1 4 -3, ––++.

1 2

3

4

Figure 6.1:�e �gure eight knot with labeled crossings.

As the reader has probably surmised, not all Gauss codes realize knots [36]. For example, the

code

1 2 -1 -2, ++

cannot belong to a knot, so we call such a code nonrealizable. In fact, a necessary but notsu�cient condition for realizability is that every number and its opposite are odd distances

apart in the Gauss word [35].

We modify the Gauss code to a code of a knot lying in the solid torus T = A × I. Let D be apunctured disk diagram. We specify a region by a subset of arcs enumerated in the direction

of the orientation, starting with the arc 0 positioned right a�er the �rst crossing we encounter

from the initial point in the direction of the orientation. An extended Gauss code is a Gausscode followed by two sequences: the arcs of the region in which the dot (i.e. "inner" component

of ∂A) of D lies on, followed by the arcs of the∞ region (the unbounded region or the "outer"component of ∂A), see Example 6.1.2. We will o�en omit the word "extended" if the variant ofthe Gauss code is evident from the context.

Example 6.1.2. �e extended Gauss code of the diagram pictured in �gure 6.2 equals

1 -2 3 -4 2 -1 4 -3, ––++, 0 3 5, 1 4 7.

We introduce a total order on the set of (extended)Gauss codes.�e order is the lexicographicalorder obtained by considering in turn:

1. the length of the Gauss code n,

2. the lexicographical ordering 1 < −1 < 2 < −2 < . . . < n < −n of the Gauss word,

Chapter 6. Classi�cation 65

1 2

3

4

0

3

1

5

7

4

2 6

Figure 6.2: A punctured disk diagram.

3. the lexicographical ordering + < − of signs,

4. the lexicographical ordering 0 < 1 < 2 < . . . < 2n − 1 of the arcs of the punctured region,

5. the lexicographical ordering 0 < 1 < 2 < . . . < 2n − 1 of the arcs of the∞ region.

IfW ′ <W and the Gauss codesW andW ′ represent the same knot, we say thatW ′ is a reduc-tion ofW . A Gauss codeW that allows a reduction is called reducible.

Example 6.1.3. �e Gauss code calculated in Example 6.1.2 is reducible, since, if we choose ourinitial point on arc 1, the Gauss code of the diagram becomes

1 -2 3 -4 2 -1 4 -3, ++––, 1 3 6, 2 5 7.

�is Gauss code is indeed a reduction of the Gauss code in Example 6.1.2, since + < − in thelexicographical ordering of signs.

6.2 Reidemeister moves

�e classi�cation algorithm uses Reidemeister moves to determine the equivalences of knots.

We overview how a Reidemeister move is performed by computer, that is, how such a trans-

formation changes the Gauss word. All equivalences below are considered up to cyclic shi�s.

Gauss subwords are denoted by A, B, and C, individual letters are denoted by i, j, and k.

An R-I move is a creation or removal of a kink:

±i ∓i A ←→ A.

An R-II move creates or removes two adjacent crossings and has either one of the following

forms:

±i ± j A ∓i ∓ j B ←→ A B,±i ± j A ∓ j ∓i B ←→ A B.

An R-III move on the Gauss word permutes the crossings involved in the move and has one of

66 6.3. �e classi�cation algorithm

the following forms:

i j A −i k B − j −k C ←→ j i A k −i B −k − j C ,i j A −i k B −k − j C ←→ j i A k −i B − j −k C ,i j A k −i B − j −k C ←→ j i A −i k B −k − j C ,i j A k −i B −k − j C ←→ j i A −i k B − j −k C .

In order to optimize the number of moves needed to �nd equivalent knots, we also add the

�ype in our collection of moves (since a �ype is relatively easy to perform on the Gauss code,

but otherwise involves a large sequence of Reidemeister moves). We only consider certain types

of �ypes – the ones where the strand enters and exits the same side of the 2-tangle T involvedin the �ype (see Figure 2.6 in Chapter 2). Such a �ype has the following e�ect on the Gauss

word:

±i W ∓i A W ′ B ←→ W A ∓iW ′±i B,

where W and W ′ are subwords consisting of letters of two types: for each letter l ∈ W (resp.

l ∈ W ′) either −l ∈ W (resp. −l ∈ W ′) or −l ∈ W ′ (resp. −l ∈ W) holds, i.e. the wordWW ′

consists only of double letters (up to signs) that correspond to crossings inside the �ipped tangle

T .�e wordW (resp.W ′) is obtained fromW (resp.W ′) by multiplying each letter by −1 (thiscorresponds to the �ip of the tangle T).

We call the isotopy that is obtained by rotating each meridional disk in the solid torus by π,a meridional rotation. We also add this operation to our collection of moves. Applying ameridional rotation on a a diagram has the e�ect of multiplying each letter in the Gauss word

by −1:

W ←→ W ,

withW being the Gauss word of the diagram.

6.3 The classification algorithm

First, we describe the algorithm that enables us to classify knots in the solid torus and continue

by describing the steps needed to classify knots in L(p, q).

6.3.1 Classifying knots in the solid torus

In order to classify knots in the solid torus T with at most n crossings, we use the followingsteps:

1. Find all possible realizable Gauss codes having length at most n.

2. Partition the codes by a common, strong enough knot invariant (the HOMFLYPT skein

module).


3. Remove Gauss codes that realize connected sums.

4. Consider the �rst knot in each partition as prime.

5. In each partition systematically perform Reidemeister moves, �ypes and meridional ro-

tations on the knots not marked as prime, until they are reduced.

We describe the steps in more detail.

Step 1. Finding all realizable Gauss codes.

Since there is a large number of Gauss codes of knot diagrams in T with at most �ve crossings(see Table 6.1), we need to optimize this step by generating only certain types of Gauss codes

that will produce all prime knots.

�e �rst optimization step is done by generating all possible Gauss codes in the lexicographical

order. If, in this process, a Gauss codeW is obviously reducible (see the next paragraph), we

can eliminateW as a Gauss code of a prime knot candidate, since the reduction ofW already

exists in the list of candidates generated beforehand.

In this �rst step of the algorithm, a Gauss code of a diagram D is considered as reducible if anycombination of the following operations reduces its code:

• renaming the letters of the Gauss code (corresponds to renaming the crossings in D),

• a cyclic shi� of the Gauss code (corresponds to choosing an initial point in D),

• reversing the Gauss code (corresponds to reversing the orientation of D),

A Gauss code is in its canonical form if it cannot be reduced using the above operations. Notethat a knot diagram can be uniquely represented by the canonical form of its Gauss code.

�e number of Gauss codes up to a length of 7 is presented in Table 6.1: the �rst row of the table

is given by the formula N = (2n)! q2n, other rows are determined experimentally.Table 6.1: Number of Gauss codes up to length 7.

n 1 2 3 4 5 6 7

Gauss codes 4 96 5760 645120 1.16 q108 3.07 q1010 1.12 q1013Realizable 4 24 432 13344 7.06 q105 5.18 q107 4.88 q109Canonical realizable 2 6 36 278 2.94 q103 3.59 q104 4.84 q105Extended realizable 36 384 10800 480384 3.46 q107 3.31 q109 3.95 q1011Canon. Ext. realizable 10 68 714 9392 1.41 q105 2.29 q106 3.91 q107

At this step we also eliminate diagrams that:

• the puncured region and the∞ region coincide (a�ne knot),

• the punctured region and the∞ region are adjacent (the sum of an a�ne knot and thenon-a�ne unknot).

Step 2. Partitioning the knots that share the common HOMFLYPT skein module (HSM).Let W be the set of all Gauss codes (i.e. diagrams of knot candidates) calculated in Step 1.

68 6.3. �e classi�cation algorithm

For eachW ∈ W , we calculate HSM and partitionW into partitions P0, P1, . . . , Pm, so that allcodes in Pi share the same HSM (i.e. are in the same equivalence class in S3(T)). HSM iscalculated recursively using the HOMFLYPT relation described in Section 2.3.2.�e recursive

algorithm also reduces the knot in each step using the breath-�rst search algorithm described

in Step 5.

Step 3. Removing Gauss codes that represent connected sums. In this step we remove allGauss codes that represent non-trivial connected sums. Note that KBSM detects connected

sums in the sense that S2,∞(K#K′) = S2,∞(K)S2,∞(K′), the proof being identical to that ofProposition 2.2.1. So, if we wish to guess if a knot K is a connected sum with, say, the a�netrefoil ♣, we check if S2,∞(K) is divisible by S2,∞(♣).

First, we calculate KBSM of allW ∈ W . If KBSM ofW ∈ W is divisible by KBSM of an a�neknot, W is a candidate for a connected sum. It turns out, in our case, that all such "divisible"

Gauss codes represented connected sums – this is veri�ed by printing out all of the connected

sum candidates and verifying that they are indeed composites by hand.

Since Step 4 is trivial, we continue with Step 5.

Step 5. Finding isotopic knot diagrams. A brute-force algorithm for �nding all possible Rei-demeister moves and search for isotopic knot diagrams would need a lot of computer power to

�nish, thus we describe the steps made to optimize this part of the algorithm.

�e key observation was already made in Step 1: we can eliminate Gauss codes that allow a re-

duction. Take each Gauss code inW not marked as prime by Step 4 and systematically performReidemeister moves, �ypes and meridional rotations up to a depth d ∈ N. We do this using abreadth-�rst search algorithm using d + 1 binary search trees (BSTs) B, B0, B1, . . . , Bd , with Bstoring all knots a�er Reidemeister moves and Bi storing knots a�er performing a Reidemeister

moves at a depth i ≤ d. A BST is a computer data structure that realizes an ordered set. Fora given Gauss code W ∈ W , the steps of the algorithm for detecting reducible codes are thefollowing:

1. InsertW into B and B0 as the root node.

2. For each i = 1, . . . , d take all codes W ′ ∈ Bi−1, systematically perform all Reidemeister

moves, and put them into their canonical form. Insert all these new Gauss codes into B,if an insertion of a code is successful, insert it also into Bi . A successful insertion of an

element into a BST means that it is a new element in the set and has not yet previously

been in it.

3. IfW ≠min(B),W has been reduced and is therefore not considered as prime.

�e advantage of using a breadth-�st algorithm is obvious: if less than d Reidemeistermoves arerequired to reduce a code, the algorithm will �nish before reaching the depth of d (as opposedto a depth-�rst algorithm that could perform d Reidemeister moves even if less than d movesare necessary). We use binary search trees for two reasons:

1. Binary search trees realize sets, so two equal codeswill never be processed simultaneously.

�is is a major step in the optimization, since, without the set structure, we could create

a kink by R-I at depth 1, remove it at depth 2, then search for the reduction of the original


knot again at depth 3. �is is the reason for inserting the Gauss code into B. �e BSTB stores all the codes appearing during the process and by checking if an insertion issuccessful, we are in fact checking if the knot has been encountered before.

2. We only use the operations of inserting elements, retrieving elements in order and re-

trieving the minimal element of a BST.�e time complexities of these operations lie in

our favor: insertion of an element into s BST is logarithmic time O(ln n), retrieving ele-ments in order and �nding the minimal element are both constant time O(1) [9].

�e other observation that improves the performance of the algorithm lies in the fact that most

of themoves R-I and R-II are redundant. For example, creating d kinks in a rowwould be point-less. Also, for a given diagram, there are in general considerably more crossing-increasing R-I

and R-II moves than there are crossing-decreasing R-I and R-II moves, R-III moves, �ypes and

meridional rotations, together – for any given arc, there are four di�erent crossing-increasing

R-I moves and for any two arcs bounding a common region, there are two di�erent crossing-

increasing R-II moves (even more if the region contains a dot). An experimentation process

on a small portion of the table was required for determining the best ratio of non-increasing

Reidemeister moves and (increasing or non-increasing) Reidemeister moves. Fortunately this

ratio turns out to be 2:1 – the best results (speed vs. successful reductions) was achieved by

making an arbitrary Reidemeister move only every third step1, at depth d ≡ 1 (mod 3).

Detecting amphichirality is simple: we mirror the knot and look in which partition the mirror

belongs: if it is in the same partition of the original knot, the knot is amphichiral, otherwise it

is chiral (note that this only works if the knots lie in partitions of size 1, otherwise, we would

have to determine amphichirality by other means).

6.3.2 Classifying knots in L(p, q)

As seen in section 3.4.1, a punctured disk diagram of a knot in L(p, q) is a punctured diskdiagram in the solid torus T accompanied by the additional slide move SL.

�e algorithm for classifying knots in L(p, q) is therefore similar to the one in T , with twodi�erences: we only search for prime knots in L(p, q) within the set of prime knots of T andwe use an L(p, q) invariant for partitioning the knots.

�us, we take the following steps in the classi�cation algorithm for L(p, q):

1. Partition all codes representing prime knots in T by a common invariant (the HOMFLYskein module for L(p, 1) and KBSM for L(p, q), q > 1).

2. Remove a�ne knots and knots that are connected sums with non-trivial a�ne knots.

3. Consider the �rst knot in each partition as prime.

4. In each partition systematically performmoves (Reidemeister moves, the SLmove, �ypes

and meridional rotations) on the knots not marked as prime, until they are reduced (if

1Perhaps it would be interesting to investigate why this is so. It may come from the observation that moving a

string through several crossings involves such a ratio of Reidemeister moves.

70 6.4. �e results

any partition is le� with more than one knot, we have to treat those knots with other

means).

All of the above steps are performed in an analogous fashion to those in the case of the solid

torus.

Unfortunately the Khovanov homology for KBSM is too cumbersome to calculate, for it being

a convenient invariant of L(2, 1). As seen in Example 5.4.3, even a 2-crossing knot requirescomplex calculations.

6.4 The results

6.4.1 Solid torus

�e number of non-a�ne prime knots in the solid torus up to n ≤ 5 crossings is presented inTable 6.2.�e knots are tabulated in Appendix A, theHOMFLYPT skeinmodules are presented

in Appendix C and the Kau�man bracket skein modules are presented in Appendix D. Since

there is a well-de�ned mirror operation in the solid torus T = A × I, i.e. re�ection of the knotthrough the annulus A× {1⁄2}, Appendix A contains only one of the possible two pairs of eachknot, the mirror of the knot is obtained by switching all undercrossings to overcrossings and

vice-versa.

Table 6.2:�e number of non-a�ne prime knots in the solid torus.

n Number of prime knots

0 1 (0 mirror pairs + 1 amphichiral)

1 2 (1 mirror pair + 0 amphichiral)




5 190⋆ (95⋆ mirror pairs + 0 amphichiral)

⋆ Assuming the Conjecture 6.4.1 is true.

Up to �ve crossings there are six non-a�ne prime amphichiral knots: 01, 22, 413, 414, 421, and 427

(see Appendix A).�ere are �ve pairs of knots that are not distinguished by KBSM: 423, 51; 51,

423; 526, 527; 526, 527; and 576, 576. �e �rst two pairs are distinguished by HSM and for the last

pair, KBSM suggests amphichirality, but HSM shows otherwise.

�e pair 526, 527 (and their mirrors 526, 527) are not distinguished neither by KBSM, neither by

HSM. Several approached have been tried to determine (in)equality of these two knots:

• a breadth-�rst search with up to 50 non-crossing-increasing Reidemeister moves, with

up to six crossing-increasing Reidemeister moves,

• a breadth-�rst searchwith up to 70Reidemeistermoves, withmaking a crossing-increasing

Reidmeister move only when no new non-crossing-increasing Reidemeister moves can


be found,

• a two-direction breadth-�rst search of the above, i.e. making the Reidemister moves of

527 and 526 simultaneously and checking if there is a non-empty intersection,

• making up to 30.000 random Reidemeister moves of the diagram 526 (and 527), reducing

the resulting diagram to a diagram with 5 crossings; this process was repeated several

thousand times with the knots not turning out to be equal,

• using several approaches, the (in)equality could not be determined from the fundamental

groups π1(T∖526) and π1(T∖527) using the computational tools GAP andMagma [15, 6].

We point out that the diagrams of knots 526 and 527 (as well as their mirrors) di�er by switch-

ing the punctured region with the unbounded region, which corresponds to the �ip operation

in [12].

An oriented knot K in a solid torus naturally corresponds to a two-component oriented linkL(K) in S3 with one trivial component – the extra trivial component being the meridian of thetorus (see Figure 6.3 for the two-component link L(526) corresponding to the knot 526) [23].Since knots 526 and 527 are so-called �ips of each other, the two corresponding links L(526) andL(527) di�er only by reversing orientations of both components of the link, i.e. it holds thatL(526) = −L(527). A link with the property that reversing both components yields the samelink is called an invertible link. We therefore transfer the question of inequivalence of links 526

and 527 (and their mirrors 526 and 527) to the question of non-invertibility of the link L(526).Non-invertibility is a very di�cult question in knot theory and only few examples of such links

have been found [59, 60].

(a) 526

∼

(b) L(526)

Figure 6.3: Diagrams of the knot 526 and the corresponding two-component link L(526).

Considering the arguments above, we state the following conjecture:

Conjecture 6.4.1. Knots 526 and 527 are not equivalent.

6.4.2 Lens spaces

We classify non-a�ne prime knots in the lens spaces L(p, q), p ≤ 12. By theorem 3.1.1, these areexactly the lens spaces L(2, 1), L(3, 1), L(4, 1), L(5, 1), L(5, 2), L(6, 1), L(7, 1), L(7, 2), L(8, 1),L(8, 3), L(9, 1), L(9, 2), L(10, 1), L(10, 3), L(11, 1), L(11, 2), L(11, 3), L(12, 1), and L(12, 5).

�ewinding numberwind(L) of a knot in L ⊂ T is the integer [L] ∈ π(T) ≅ Z, or equivalently,the integer [D] ∈ π(R2 ∖ { q }) ≅ Z if D is a diagram of L lying in the punctured plane R2 ∖{ q}.


Proposition 6.4.1. Let i ∶ T ↪ L(p, q) be the standard inclusion of the solid torus T in the lensspace L(p, q). If two knots K1 and K2 with at most n crossings have distinct KBSM in the solidtorus, then the knots i(K1) and i(K2) have distinct KBSM in L(p, q), p ≥ 2(n + 1).

Proof. Since a knot K ⊂ T that allows a diagram D with n crossings has the winding numberbounded by ∣wind(K)∣ ≤ n+1 (D can go up to (n+1) times around the puncture and thus needsn crossings to complete the circle), it can be expressed in the standard generators of KBSM ofthe solid torus as an expression S2,∞(T)(K) = ∑n+1

i=0 Aix i , for some Ai ∈ R. Since p ≥ 2(n+1)⇒⌊p/2⌋ ≥ n + 1 and the generators {x′n}⌊p/2⌋

n=0 of KBSM of L(p, q) are induced by the inclusion,i∗(x i) = x′i , 0 ≤ i ≤ ⌊p/2⌋, it holds that S2,∞(L(p, q))(i(K)) = ∑n+1

i=0 Aix′i , Ai ∈ R.�at is, theknots K and i(K), although lying in di�erent spaces, have an equal KBSM expression.

A similar proposition can be made for the HOMFLYPT skein module.

Proposition 6.4.2. Let i ∶ T ↪ L(p, 1) be the standard inclusion of the solid torus T in the lensspace L(p, 1). If two knots K1 and K2 with at most n crossings have distinct HSM in the solid torus,then the knots i(K1) and i(K2) have distinct HSM in L(p, 1), p > 2(n + 1).

Proof. Since the generators {t i1k1 . . . tisks ∶ s ∈ N, k1 < .. < ks ∈ Z ∖ {0},− p

2< k1 < .. < ks ≤ p

2, i1..is ∈

N}∪ {∅} of HSM of L(p, 1) are induced by the inclusion of the generators of HSM of T , by thesame arguments that we use in Proposition 6.4.1 and by the fact that p > 2(n + 1)⇒ p

2> n, we

conclude that the expression S3(T)(K) in the standard generators is equal to the expression ofS3(L(p, 1))(i(K)).

From Conjecture 4.4.1 we have the following corollary:

Conjecture 6.4.2. Let i ∶ T ↪ L(p, q) be the standard inclusion of the solid torus T in the lensspace L(p, q), q > 1. If two knots K1 and K2 with at most n crossings have distinct HSM in thesolid torus, then the knots i(K1) and i(K2) have distinct HSM in L(p, q), p > 2(n + 1), q > 1.

�e number of non-a�ne prime knots with up to �ve crossings in L(p, q) is presented in Ta-ble 6.3. �e set of knots that are either reducible, a�ne or composites in L(p, q) are tabulatedin Appendix B, i.e. the classi�cation is given by the set of prime knots in the solid torus minus

the set of knots in Appendix B. Note that we also tabulate the mirrors (in terms of mirrors in

the solid torus), since there is no well-de�ned mirror operation in the punctured disk diagram

of L(p, q), e.g. the knot 11 is the unknot in L(2, 1), but the knot 11 is not.

Since KBSM is unique for all knots up to four crossings in the solid torus and the lens spaces

L(p, q), p < 10, we conclude by specializing Proposition 6.4.1 to the case n = 4, that KBSM dis-tinguishes these knots in all lens spaces. A similar argument can be made for the HOMFLYPT

skein module: all knots up to four crossings are unique in HSM of the solid torus and L(p, 1),p ≤ 10, so they are unique in HSM of all L(p, 1).

�ere are several irreducible knots that are excluded from the tables:

• In L(2, 1): the knot 577 is a connected sumwith the trefoil, the knot 524 is the a�ne trefoil,and the knot 11 is the a�ne unknot.

• In L(3, 1): the knot 23 is the a�ne unknot.


Table 6.3: Number of non-a�ne prime knots in L(p, q) with up ot 5 crossings.n L(2, 1) L(3, 1) L(4, 1) L(5, 1) L(5, 2) L(6, 1) L(7, 1)0 1 1 1 1 1 1 1

1 1 1 2 2 2 2 2

2 4 4 5 4 4 5 5

3 12 13 14 15 16 15 15

4 40 42 45 46 49 48 49

5 155⋆ 163⋆ 176⋆ 183⋆ 189⋆◇ 184⋆ 186⋆

n L(7, 2) L(8, 1) L(8, 3) L(9, 1) L(9, 2) L(10, 1) L(10, 3)0 1 1 1 1 1 1 1

1 2 2 2 2 2 2 2

2 5 5 5 5 5 5 5

3 15 16 16 16 16 16 16

4 50 49 50 49 49 50 50

5 189⋆◇ 188⋆ 190⋆◇ 189⋆ 190⋆◇ 189⋆ 190⋆◇

n L(11, 1) L(11, 2) L(11, 3) L(12, 1) L(12, 5) L(p, q), p ≥ 130 1 1 1 1 1 1

1 2 2 2 2 2 2

2 5 5 5 5 5 5

3 16 16 16 16 16 16

4 50 50 50 50 50 50

5 189⋆ 189⋆◇ 190⋆◇ 190⋆ 190⋆◇ 190⋆◇

⋆ Assuming Conjecture 6.4.3 is true.◇ Assuming Conjecture 6.4.4 is true.




In Table 6.4 we present inequivalent knots that are not distinguished by KBSM, for p = 1 all butthe pair 526, 527 (and their mirrors) are distinguishable by HSM. For p > 1 we have to conjectureabout the knots in the solid torus that are distinguished only by HSM. Recall that these are

exactly the pairs 423, 51; 526, 527; and 576, 576 (and the mirrors of the �rst two pairs).

A high depth breadth-�rst searched has been performed to check that these knots are (plausibly)

not equivalent in L(p, q), p ≤ 12.

Conjecture 6.4.3. Knots 526 and 527 and their solid torus mirrors are not equivalent in anyL(p, q).

Conjecture 6.4.4. �e pair 423, 51 (and their mirrors); and the pair 576, 576 are not equivalent inL(p, q), q > 1.

We point out that the Conjecture 6.4.4 is a corollary of Conjecture 4.4.1.


We also point out that in Table 6.4 the conjecture above also applies to the case of L(5, 2) sincethe knot 36 is a reduction of the knot 423 and the knot 575 is a reduction of the knot 576 (that is,

it holds that 36 = 423 and 575 = 576 in L(5, 2), see Appendix B).


Table 6.4: Ineqivalent prime knots in L(p, q) that share KBSM.Space Knots that share KBSML(2, 1) 23, 51, 55, 576; 41, 410; 43, 411; 48, 416;

423, 51; 510, 532; 526, 5⋆27; 526, 527

⋆; 531, 560;

536, 560; 539, 552

L(3, 1) 11, 51; 21, 576; 35, 49; 423, 51; 517, 528;

511, 559; 526, 5⋆27; 526, 527

⋆; 549, 556

L(4, 1) 22, 51; 34, 576; 46, 577; 423, 51; 526, 5⋆27;

526, 527⋆; 549, 595

L(5, 1) 32, 51; 417, 576; 423, 51; 526, 5⋆27; 526, 527

⋆

L(5, 2) 36, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 575, 576

◇

L(6, 1) 42, 51; 418, 53; 423, 51, 572; 526, 5⋆27; 526, 527

⋆;

568, 576

L(7, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(7, 2) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(8, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(8, 3) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(9, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(9, 2) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(10, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(10, 3) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(11, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(11, 2) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(11, 3) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(12, 1) 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(12, 5) 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

L(p, 1)† 423, 51; 423, 51; 526, 5⋆27; 526, 527

⋆; 576, 576

L(p, q)‡ 423, 51◇; 423, 5

◇1 ; 526, 5

⋆27; 526, 527

⋆; 576, 576

◇

⋆ Assuming Conjecture 6.4.3 is true.◇ Assuming Conjecture 6.4.4 is true.† p > 12.‡ p > 12, q > 1.

7Conclusion and open questions

�e HOMFLYPT skein modules of L(p, 1) have been calculated. We have shown that they arefree R-modules generated by an in�nite set of generators.

Clearly the open question from here on lies in calculating the HOMFLYPT skein modules of

L(p, q), q > 1. We have given a conjecture that they are free R-modules with an in�nite set ofgenerators.�e next open question, which, by the author’s opinion, can be calculated by similar

methods of those used in this thesis, is the calculation of the 2-variable Kau�man skein module

of L(p, q).We have categori�ed the Kau�man bracket skeinmodule (KBSM) of the projective space.

With this result, we closed the chapter of categorifying the Kau�man bracket skein module of

links in I-bundles over surfaces (since this was the only remaining case le� in [3]) and per-haps opened a new chapter on categorifying Jones-like invariants of other 3-manifolds, the

most obvious candidates being the general lens space L(p, q) and S1-bundles over surfaces.For the case of L(p, q), the main di�culty of categorifying KBSM lies in the question of devel-oping a non-recursive state-sum formula for describing a link in the basis of the skein module.

�e author’s opinion is, that no easy-to-describe formula exists for the KBSM of L(p, q) with(p, q) ≠ (2, 1).Up to now, knots have been classi�ed only in the 3-dimensional Euclidean space, the projective

space, and, up to a so called "�ip", in the solid torus. We have re�ned the classi�cation in the

solid torus by giving a full classi�cation of knots up to four crossings. We have also given a

classi�cation up to �ve crossings with one exception: we have not been able to prove that the

knots 526 and 527 are inequivalent, but have provided evidence that this is probably the case.

We have given a full classi�cation of knots in all lens spaces L(p, q) up to four crossings and aclassi�cation up to �ve crossings with a few exceptions.

Preliminary calculations show that, using the same techniques that we have used, a few dozen

of similar exceptions appear in the classi�cation up to six crossings, both in the solid torus and

in L(p, q). Developing techniques that will resolve these di�culties and a classi�cation of knotsin L(p, q) with a higher number of crossings is certainly a result the author wishes to achievein the near future.

77

Appendices

79

ATable of knots in the solid torus

01 11 21∗22 23 31

32 33 34 35 36 37

38 41 42 43 44 45

46 47 48 49 410 411

81

82

412∗413

∗414 415 416

417 418 419 420∗421

422 423 424 425 426

∗427 51 52 53 54

55 56 57 58 59

510 511 512 513 514

Appendix A. Table of knots in the solid torus 83

515 516 517 518 519

520 521 522 523 524

525 526 527 528 529

530 531 532 533 534

535 536 537 538 539

540 541 542 543 544

84

545 546 547 548 549

550 551 552 553 554

555 556 557 558 559

560 561 562 563 564

565 566 567 568 569

570 571 572 573 574

Appendix A. Table of knots in the solid torus 85

575 576 577 578 579

580 581 582 583 584

585 586 587 588 589

590 591 592 593 594

595

Legend: ∗ – amphichiral knot

BEquivalences of knots in lens spaces

Table B.1: Equivalences of knots in lens spaces.

Space EquivalencesL(2, 1) 11, 574 ∼ O 416 ∼ 34 52 ∼ 46 524 ∼ ♣

36, 38 ∼ 11 541, 590 ∼ 33 427 ∼ 47 554 ∼ 532

37 ∼ 21 585 ∼ 36 587 ∼ 410 551, 592 ∼ 532

426, 575, 581 ∼ 21 41, 588 ∼ 38 539 ∼ 411 587 ∼ 543

422, 423, 576, 578 ∼ 23 422 ∼ 42 425 ∼ 417 578 ∼ 568

424 ∼ 23 593 ∼ 46 589 ∼ 420 577 ∼ 01#♣37, 594 ∼ 32 542 ∼ 47 55 ∼ 423 591 ∼ 579

425, 571 ∼ 33 54, 591 ∼ 49 538, 567 ∼ 426 593 ∼ 580

595 ∼ 35 53, 523 ∼ 48 574 ∼ 56

L(3, 1) 36, 574, 575 ∼ 11 578, 581 ∼ 33 52, 577 ∼ 47 54 ∼ 53

423, 424, 524 ∼ 11 49 ∼ 35 529 ∼ 415 533, 582 ∼ 522

422 ∼ 21 426 ∼ 34 590 ∼ 411 541 ∼ 523

38 ∼ 22 55 ∼ 38 425 ∼ 416 555 ∼ 534

576 ∼ 21 42 ∼ 37 567 ∼ 417 591 ∼ 539

23, 571 ∼ O 420 ∼ 46 56 ∼ 422 593 ∼ 540

585 ∼ 23 588 ∼ 47 568 ∼ 425 587 ∼ 542

37 ∼ 31 594 ∼ 48 538 ∼ 427

L(4, 1) 575, 585 ∼ 11 38 ∼ 32 578 ∼ 416 577 ∼ 538

574 ∼ 21 576 ∼ 34 539 ∼ 418 572 ∼ 540

423 ∼ 22 36 ∼ O 426 ∼ 417 590 ∼ 543

424 ∼ 23 41 ∼ 37 567 ∼ 425 588 ∼ 579

422 ∼ 31 56 ∼ 37 571 ∼ 524 594 ∼ 580

87

88

Space EquivalencesL(5, 1) 23 ∼ 11 423 ∼ 32 42 ∼ 38 424 ∼ O

575 ∼ 22 585 ∼ 36 422 ∼ 41 568 ∼ 426

574 ∼ 31 55 ∼ 37 576 ∼ 417 578 ∼ 567

L(5, 2) 23 ∼ 11 423 ∼ 36 576 ∼ 575

L(6, 1) 36 ∼ 11 56 ∼ 38 423 ∼ 42 576 ∼ 568

575 ∼ 32 574 ∼ 41 55 ∼ 422 585 ∼ OL(7, 1) 424 ∼ 11 575 ∼ 42 574 ∼ 55

36 ∼ 23 56 ∼ 423

L(7, 2) 36 ∼ 23 575 ∼ 424

L(8, 1) 585 ∼ 11 424 ∼ 23 575 ∼ 56

L(8, 3) /

L(9, 1) 585 ∼ 23 424 ∼ 36

L(9, 2) 424 ∼ 36

L(10, 1) 585 ∼ 36

L(10, 3) /

L(11, 1) 585 ∼ 424

L(11, 2) 585 ∼ 424

L(11, 3) /

L(12, 1) /

L(12, 5) /

Legend: O – a�ne unknot♣ – a�ne trefoil

CThe HOMFLYPT skein modules

�e HOMFLYPT skein modules of the non-a�ne prime knots are expressed in the standard

basis using the evaluation ⟨∅⟩ = 1, see�eorem 4.1.1 and�eorem 4.2.1. For knots in L(p, 1) thatare not listed below (and all knots in L(12, 1)) the HSM expressions equal the HSM expressionsof the corresponding knots in the solid torus.

Knots in the solid torus

S3(01) = t1S3(11) = t2S3(11) = v−2t2 − v−1zt21S3(21) = vzt−1t1 − v3z−1 + vz−1

S3(21) = −v−1zt−1t1 − v−1z−1 + v−3z−1

S3(22) = −v−1zt1t2 + v−2t3S3(23) = t3S3(23) = v−2z2t31 − 2v−3zt1t2 + v−4t3S3(31) = vzt−1t2 + v2t1S3(31) = v−2z2t−1t21 − v−3zt−1t2 + v−2t1S3(32) = v−2z2t21 t2 − v−3zt1t3 − v−3zt22 + v−4t4S3(32) = −v−1zt1t3 + v−2t4S3(33) = vzt21 + v2t2S3(33) = −v−1zt21 − v−3zt21 + v−4t2S3(34) = −z2t−1t21 + v−1zt−1t2 + v2t1S3(34) = −v−1zt−1t2 + v−2t1S3(35) = −v−1zt22 + v−2t4S3(35) = v−2z2t21 t2 − 2v−3zt1t3 + v−4t4S3(36) = t4S3(36) = −v−3z3t41 + 3v−4z2t21 t2 − 2v−5zt1t3 − v−5zt22 + v−6t4S3(37) = v3zt21 + v2z2t2 + v2t2S3(37) = −v−3z3t21 − 2v−3zt21 + v−4z2t2 + v−4t2S3(38) = v3zt−1t1 − v3z − v3z−1 + vz + vz−1

S3(38) = −v−3zt−1t1 − v−1z − v−1z−1 + v−3z + v−3z−1

89

90 Knots in the solid torus

S3(41) = vzt−1t3 + v2t2S3(41) = −v−3z3t−1t31 + 2v−4z2t−1t1t2 − v−5zt−1t3 − v−3zt21 + v−4t2S3(42) = −v−3z3t31 t2 + v−4z2t21 t3 + 2v−4z2t1t22 − v−5zt1t4 − 2v−5zt2t3 + v−6t5S3(42) = −v−1zt1t4 + v−2t5S3(43) = v2z2t−1t21 − v4t1 + 2v2t1S3(43) = v−2z2t−1t21 + 2v−2t1 − v−4t1S3(44) = −z2t−1t1t2 + v−1zt−1t3 − vzt21 + t2S3(44) = v−2z2t−1t1t2 − v−3zt−1t3 + v−2t2S3(45) = v−2z2t1t22 − 2v−3zt2t3 + v−4t5S3(45) = v−2z2t21 t3 − 2v−3zt1t4 + v−4t5S3(46) = v3z3t−1t1 + 2v3zt−1t1 − v5z − v5z−1 + v3z + v3z−1

S3(46) = −v−3z3t−1t1 − 2v−3zt−1t1 − v−3z − v−3z−1 + v−5z + v−5z−1

S3(47) = v3zt−1t2 + v2z2t1 + v2t1S3(47) = v−4z2t−1t21 − v−5zt−1t2 + v−2z2t1 + v−2t1S3(48) = −v−3z3t1t2 − 2v−3zt1t2 + v−4z2t3 + v−4t3S3(48) = −v2z2t31 − vz3t1t2 + z2t3 + t3S3(49) = −v−3zt21 + v−2z2t2 + v−2t2S3(49) = v3zt21 − vz3t21 − vzt21 + z2t2 + t2S3(410) = v3zt−1t1 + vzt−1t1 − v5z−1 + v3z−1

S3(410) = −v−1zt−1t1 − v−3zt−1t1 − v−3z−1 + v−5z−1

S3(411) = −z2t31 − vzt1t2 + v−1zt1t2 + t3S3(411) = −v−1zt1t2 − v−3zt1t2 + v−4t3S3(412) = −z2t−1t1t2 + v−1zt−1t3 + v2t2S3(412) = v−2z2t−1t1t2 − v−3zt−1t3 − v−3zt21 + v−4t2S3(413) = −z2t−1t21 + v2t1 − t1 + v−2t1S3(414) = v−2z2t1t22 − v−3zt1t4 − v−3zt2t3 + v−4t5S3(415) = −v−1zt−2t2 + v−1zt−1t1 − vz−1 + v−1z−1

S3(415) = −v−2z2t−2t21 + v−3zt−2t2 + v−1z3t2−1t21 − v−2z2t2−1t2 − vzt−1t1 − vz−1 + v−1z−1

S3(416) = vzt1t2 + v2t3S3(416) = v−2z2t31 + v−4z2t31 − v−3zt1t2 − 2v−5zt1t2 + v−6t3S3(417) = v−1z3t−1t31 − 2v−2z2t−1t1t2 + v−3zt−1t3 − vzt21 + t2S3(417) = −v−1zt−1t3 + v−2t2S3(418) = −v−1zt2t3 + v−2t5S3(418) = −v−3z3t31 t2 + 2v−4z2t21 t3 + v−4z2t1t22 − 2v−5zt1t4 − v−5zt2t3 + v−6t5S3(419) = v−2z2t−2t21 − v−3zt−2t2 − v−1z−1 + v−3z−1

S3(419) = −z2t−2t21 + v−1zt−2t2 − v3z−1 + vz−1

S3(420) = vzt−2t2 + v3zt−1t1 − v5z−1 + v3z−1

S3(420) = v−4z2t−2t21 − v−5zt−2t2 − v−3z3t2−1t21 + v−4z2t2−1t2 − v−3zt−1t1 − v−3z−1 + v−5z−1

S3(421) = v−2z2t21 t3 − v−3zt1t4 − v−3zt2t3 + v−4t5S3(422) = v3zt1t2 + v2z2t3 + v2t3S3(422) = v−4z4t31 + 2v−4z2t31 − 2v−5z3t1t2 − 3v−5zt1t2 + v−6z2t3 + v−6t3S3(423) = −v−3zt−1t2 + v−2z2t1 + v−2t1S3(423) = −v2z2t−1t21 + vzt−1t2 + v2z2t1 + v2t1S3(424) = t5S3(424) = v−4z4t51 − 4v−5z3t31 t2 + 3v−6z2t21 t3 + 3v−6z2t1t22 − 2v−7zt1t4 − 2v−7zt2t3 + v−8t5

Appendix C. �e HOMFLYPT skein modules 91

S3(425) = v−1zt−1t1 − v3z−1 + vz + vz−1 − v−1zS3(425) = −vzt−1t1 + vz − v−1z − v−1z−1 + v−3z−1

S3(426) = −vzt21 − z2t2 + v−2t2S3(426) = −vzt21 + v−1z3t21 + v−1zt21 + t2 − v−2z2t2S3(427) = z2t31 + v−1z3t1t2 − v−1zt1t2 − v−2z2t3 + v−2t3S3(51) = v4z2t−1t21 + v3z3t−1t2 + v3zt−1t2 − v6t1 + v4z2t1 + 2v4t1S3(51) = v−4z4t−1t21 + 2v−4z2t−1t21 − v−5z3t−1t2 − v−5zt−1t2 + v−4z2t1 + 2v−4t1 − v−6t1S3(52) = −v2z2t−1t1t2 + vzt−1t3 + v2z2t2 + v2t2S3(52) = v−4z2t−1t1t2 − v−5zt−1t3 − v−3z3t21 − v−3zt21 + v−4z2t2 + v−4t2S3(53) = −v−2z2t−1t21 + t1 + v−2z2t1S3(53) = −v2z2t−1t21 + v2z2t1 + t1S3(54) = v−4z4t21 t2 + 2v−4z2t21 t2 − v−5z3t1t3 − 2v−5zt1t3 − v−5z3t22 − v−5zt22 + v−6z2t4 + v−6t4S3(54) = −v2z2t21 t2 − vz3t1t3 + z2t4 + t4S3(55) = vzt−1t4 + v2t3S3(55) = v−4z4t−1t41 − 3v−5z3t−1t21 t2 + 2v−6z2t−1t1t3 + v−6z2t−1t22 − v−7zt−1t4 + v−4z2t31 − 2v−5zt1t2 + v−6t3S3(56) = v−4z4t41 t2 − v−5z3t31 t3 − 3v−5z3t21 t22 + v−6z2t21 t4 + 4v−6z2t1t2t3 − v−7zt1t5 + v−6z2t32 − 2v−7zt2t4 −v−7zt23 + v−8t6S3(56) = −v−1zt1t5 + v−2t6S3(57) = v2z2t2−1t2 + 2v3zt−1t1 − v5z−1 + v3z−1

S3(57) = v−4z2t−2t21 − v−3z3t2−1t21 − 2v−3zt−1t1 − v−3z−1 + v−5z−1

S3(58) = −z2t−1t1t3 + v−1zt−1t4 − vzt1t2 + t3S3(58) = −v−3z3t−1t21 t2 + v−4z2t−1t1t3 + v−4z2t−1t22 − v−5zt−1t4 − v−3zt1t2 + v−4t3S3(59) = −v−3z3t21 t22 + 2v−4z2t1t2t3 + v−4z2t32 − 2v−5zt2t4 − v−5zt23 + v−6t6S3(59) = v−2z2t21 t4 − 2v−3zt1t5 + v−4t6S3(510) = v2z2t−1t21 + v3zt−1t2 + v2t1S3(510) = v−2z2t−1t21 + v−4z2t−1t21 − v−5zt−1t2 + v−2t1S3(511) = −vz3t−1t31 + z2t−1t1t2 + v3zt21 − vzt21 + t2S3(511) = v−2z2t−1t1t2 − v−3zt21 + v−2t2S3(512) = −z2t−1t22 + v−1zt−1t4 − vzt1t2 + t3S3(512) = −v−3z3t−1t21 t2 + 2v−4z2t−1t1t3 − v−5zt−1t4 − v−3zt1t2 + v−4t3S3(513) = −z2t2−1t2 − vzt−1t1 + v−1zt−1t1 − vz−1 + v−1z−1

S3(513) = −v−2z2t−2t21 + v−1z3t2−1t21 − vzt−1t1 + v−1zt−1t1 − vz−1 + v−1z−1

S3(514) = v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 + v−3zt−1t4 − vzt1t2 + t3S3(514) = v−2z2t−1t1t3 − v−3zt−1t4 − v−3zt1t2 + v−4t3S3(515) = −v−3z3t21 t22 + 3v−4z2t1t2t3 − v−5zt1t5 − v−5zt2t4 − v−5zt23 + v−6t6S3(515) = v−2z2t1t2t3 − v−3zt1t5 − v−3zt2t4 + v−4t6S3(516) = v−2z2t−2t1t2 − v−3zt−2t3 + v−2t1S3(516) = −v−2z2t−2t1t2 + v−3zt−2t3 + v−1z3t2−1t1t2 − v−2z2t2−1t3 + v2t1S3(517) = v−2z2t21 t2 + v−4z2t21 t2 − v−3zt1t3 − v−5zt1t3 − v−5zt22 + v−6t4S3(517) = −z2t21 t2 − vzt1t3 + v−1zt1t3 + t4S3(518) = v2z2t−1t1t2 + v3zt21 + v2t2S3(518) = −v−3z3t−1t31 + v−4z2t−1t1t2 − 2v−3zt21 + v−4t2S3(519) = v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 + v−3zt−1t4 + z2t31 − 2v−1zt1t2 + v−2t3S3(519) = v−2z2t−1t1t3 − v−3zt−1t4 + v−2t3S3(520) = v−2z2t1t2t3 − v−3zt2t4 − v−3zt23 + v−4t6


S3(520) = −v−3z3t31 t3 + 2v−4z2t21 t4 + v−4z2t1t2t3 − 2v−5zt1t5 − v−5zt2t4 + v−6t6S3(521) = v−4z2t−2t1t2 − v−5zt−2t3 − v−3z3t2−1t1t2 + v−4z2t2−1t3 − v−3zt−1t2 + v−4t1S3(521) = −z2t−2t1t2 + v−1zt−2t3 − v2z2t−1t21 + vzt−1t2 + v4t1S3(522) = −v−1zt−1t3 + v−1zt21 + t2S3(522) = v−1z3t−1t31 − 2v−2z2t−1t1t2 + v−3zt−1t3 − vzt21 − v−1zt21 + v−2t2S3(523) = −v−2z2t−1t21 + v−3zt−1t2 + 2t1 − v−2t1S3(523) = −vzt−1t2 − v2t1 + 2t1S3(524) = vzt1t3 − vzt22 + t4S3(524) = −v−3z3t41 + 3v−4z2t21 t2 − 3v−5zt1t3 + v−6t4S3(525) = v3zt−1t3 + v3z3t21 + v3zt21 + v4z2t2 + v4t2S3(525) = −v−5z3t−1t31 + 2v−6z2t−1t1t2 − v−7zt−1t3 − v−3z3t21 − v−3zt21 − v−5z3t21 − v−5zt21 + v−6z2t2 + v−6t2S3(526) = −v2z4t−1t21 − v2z2t−1t21 + vz3t−1t2 + vzt−1t2 + v4z2t1 + v2t1S3(526) = −v−3z3t−1t2 − v−3zt−1t2 + v−2t1 + v−4z2t1S3(527) = −v2z4t−1t21 − v2z2t−1t21 + vz3t−1t2 + vzt−1t2 + v4z2t1 + v2t1S3(527) = −v−3z3t−1t2 − v−3zt−1t2 + v−2t1 + v−4z2t1S3(528) = −v−3zt1t3 − v−3z3t22 − v−3zt22 + v−4z2t4 + v−4t4S3(528) = vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − v−1zt1t3 + v−2z2t4 + v−2t4S3(529) = −vz3t−1t1 − vzt−1t1 + v−1zt−1t1 + v3z − vz − vz−1 + v−1z−1

S3(529) = −vzt−1t1 + v−1z3t−1t1 + v−1zt−1t1 − vz−1 + v−1z + v−1z−1 − v−3zS3(530) = −vzt−1t2 − z2t1 + v−2t1S3(530) = −v−2z2t−1t21 + v−3zt−1t2 + v2t1 − z2t1S3(531) = z4t−1t21 + z2t−1t21 − v−1z3t−1t2 − v2z2t1 + t1S3(531) = z2t−1t21 + v−1z3t−1t2 + t1 − v−2z2t1S3(532) = −vzt1t2 + v−1z3t1t2 + v−1zt1t2 + t3 − v−2z2t3S3(532) = z2t31 + v−1z3t1t2 − v−1zt1t2 − v−3zt1t2 − v−2z2t3 + v−4t3S3(533) = v−1zt21 + v2t2 − z2t2S3(533) = −vzt21 + v−1z3t21 − v−3zt21 − v−2z2t2 + v−4t2S3(534) = z2t21 t2 − v−1zt1t3 + v−1z3t22 − v−2z2t4 + v−2t4S3(534) = −v−1z3t41 − v−2z4t21 t2 + 2v−2z2t21 t2 + 2v−3z3t1t3 − v−3zt1t3 − v−3zt22 − v−4z2t4 + v−4t4S3(535) = v3zt−2t2 + v3z3t−1t1 + v3zt−1t1 − v5z − v5z−1 + v3z + v3z−1

S3(535) = v−6z2t−2t21 −v−7zt−2t2−v−5z3t2−1t21 +v−6z2t2−1t2−v−3z3t−1t1−v−3zt−1t1−v−3z−v−3z−1+v−5z+v−5z−1

S3(536) = v3zt1t2 − vz3t1t2 − vzt1t2 + z2t3 + t3S3(536) = v−4z2t31 − v−3z3t1t2 − v−3zt1t2 − v−5zt1t2 + v−4z2t3 + v−4t3S3(537) = v−1z3t−1t1 + v−1zt−1t1 − v−3zt−1t1 − vz − vz−1 + v−1z + v−1z−1

S3(537) = v3zt−1t1 − vz3t−1t1 − vzt−1t1 − vz − vz−1 + v−1z + v−1z−1

S3(538) = −v2z2t−2t21 + vzt−2t2 − v3z − v3z−1 + vz + vz−1

S3(538) = v−4z2t−2t21 − v−5zt−2t2 − v−1z − v−1z−1 + v−3z + v−3z−1

S3(539) = −v−3zt1t2 + v−2z2t3 + v−2t3S3(539) = −v2z2t31 + z4t31 + z2t31 + vzt1t2 − 2v−1z3t1t2 − 2v−1zt1t2 + v−2z2t3 + v−2t3S3(540) = v3z3t−1t2 + 2v3zt−1t2 + v4z2t1 + v4t1S3(540) = v−4z4t−1t21 + 2v−4z2t−1t21 − v−5z3t−1t2 − 2v−5zt−1t2 + v−4z2t1 + v−4t1S3(541) = v−4z4t21 t2 + 2v−4z2t21 t2 − v−5z3t1t3 − v−5zt1t3 − v−5z3t22 − 2v−5zt22 + v−6z2t4 + v−6t4S3(541) = −v2z2t21 t2 − vz3t1t3 − vzt1t3 + vzt22 + z2t4 + t4S3(542) = v3zt−1t2 + vzt−1t2 + v4t1


S3(542) = v−2z2t−1t21 + v−4z2t−1t21 − v−3zt−1t2 − v−5zt−1t2 + v−4t1S3(543) = v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − v−1zt1t3 + v−3zt1t3 − v−1zt22 + v−2t4S3(543) = −v−1zt1t3 − v−3zt1t3 + v−4t4S3(544) = −z2t−1t1t3 + v−1zt−1t4 + v2t3S3(544) = −v−3z3t−1t21 t2 + v−4z2t−1t1t3 + v−4z2t−1t22 − v−5zt−1t4 + v−4z2t31 − 2v−5zt1t2 + v−6t3S3(545) = v−1z3t−1t31 − v−2z2t−1t1t2 − vzt21 + v−1zt21 + t2S3(545) = −z2t−1t1t2 − vzt21 + v−2t2S3(546) = −z2t−2t1t2 + v−1zt−2t3 + v2t1S3(546) = v−4z2t−2t1t2 − v−5zt−2t3 − v−3z3t2−1t1t2 + v−4z2t2−1t3 + v−2t1S3(547) = −v−3z3t21 t22 + v−4z2t21 t4 + v−4z2t1t2t3 − v−5zt1t5 + v−4z2t32 − 2v−5zt2t4 + v−6t6S3(547) = v−2z2t1t2t3 − v−3zt1t5 − v−3zt23 + v−4t6S3(548) = −v−1zt−2t3 + v−1zt−1t2 + t1S3(548) = v−3z3t−2t31 − 2v−4z2t−2t1t2 + v−5zt−2t3 − v−2z4t2−1t31 + 2v−3z3t2−1t1t2 − v−4z2t2−1t3 + z2t−1t21 −v−1zt−1t2 + t1S3(549) = v−2z2t21 t2 + v−4z2t21 t2 − v−5zt1t3 − v−3zt22 − v−5zt22 + v−6t4S3(549) = −z2t21 t2 − vzt1t3 + v−1zt22 + t4S3(550) = v3z3t21 + 2v3zt21 + v4z2t2 + v4t2S3(550) = −v−3z3t21 − 2v−3zt21 − v−5z3t21 − v−5zt21 + v−6z2t2 + v−6t2S3(551) = v4z2t−1t21 − v6t1 + v4t1 + v2z2t1 + v2t1S3(551) = v−4z2t−1t21 + v−2z2t1 + v−2t1 + v−4t1 − v−6t1S3(552) = −v−2z4t−1t21 − 2v−2z2t−1t21 + v−3z3t−1t2 + v−3zt−1t2 + z2t1 + 2t1 − v−2t1S3(552) = −v2z2t−1t21 − vz3t−1t2 − vzt−1t2 − v2t1 + z2t1 + 2t1S3(553) = −v2z2t21 t2 + vzt1t3 − vz3t22 − vzt22 + z2t4 + t4S3(553) = v−4z4t21 t2 + 2v−4z2t21 t2 − 2v−5z3t1t3 − 3v−5zt1t3 + v−6z2t4 + v−6t4S3(554) = −v−3z3t−1t2 − 2v−3zt−1t2 + v−4z2t1 + v−4t1S3(554) = −v2z4t−1t21 − 2v2z2t−1t21 + vz3t−1t2 + 2vzt−1t2 + v4z2t1 + v4t1S3(555) = v−4z2t−1t1t2 − v−5zt−1t3 + v−2z2t2 + v−2t2S3(555) = −v2z2t−1t1t2 + vzt−1t3 − vz3t21 − vzt21 + z2t2 + t2S3(556) = −v−3z3t22 − 2v−3zt22 + v−4z2t4 + v−4t4S3(556) = vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − 2v−1zt1t3 + v−1zt22 + v−2z2t4 + v−2t4S3(557) = v3zt21 + vzt21 + v4t2S3(557) = −v−1zt21 − v−3zt21 − v−5zt21 + v−6t2S3(558) = −v2z2t−1t21 − z2t−1t21 + vzt−1t2 + v−1zt−1t2 + v4t1S3(558) = −v−1zt−1t2 − v−3zt−1t2 + v−4t1S3(559) = −z2t21 t2 + v−1zt1t3 − vzt22 + t4S3(559) = v−2z2t21 t2 + v−4z2t21 t2 − v−3zt1t3 − 2v−5zt1t3 + v−6t4S3(560) = −z2t−1t21 − vzt−1t2 + v−2t1S3(560) = −z2t−1t21 − v−2z2t−1t21 + v−3zt−1t2 + v2t1S3(561) = −vz3t2−1t21 + z2t2−1t2 + 2v3zt−1t1 − v5z−1 + v3z−1

S3(561) = v−2z2t−2t21 − 2v−3zt−1t1 − v−3z−1 + v−5z−1

S3(562) = v−1z3t−1t21 t2 − 2v−2z2t−1t1t3 + v−3zt−1t4 − vzt1t2 + t3S3(562) = v−2z2t−1t22 − v−3zt−1t4 − v−3zt1t2 + v−4t3S3(563) = −z2t−1t1t2 + v−1zt21 + v2t2S3(563) = v−1z3t−1t31 − v−2z2t−1t1t2 − vzt21 − v−3zt21 + v−4t2S3(564) = −v−2z2t−2t1t2 + v−3zt−2t3 + v−1z3t2−1t1t2 − v−2z2t2−1t3 − vzt−1t2 + t1


S3(564) = v−2z2t−2t1t2 − v−3zt−2t3 − v−2z2t−1t21 + v−3zt−1t2 + t1S3(565) = v−2z2t32 − 2v−3zt2t4 + v−4t6S3(565) = −v−3z3t21 t22 + v−4z2t21 t4 + 2v−4z2t1t2t3 − 2v−5zt1t5 − v−5zt23 + v−6t6S3(566) = −v−1zt22 − v−3zt22 + v−4t4S3(566) = v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − 2v−1zt1t3 + v−3zt22 + v−2t4S3(567) = vzt1t3 + v2t4S3(567) = −v−3z3t41 − v−5z3t41 + 2v−4z2t21 t2 + 3v−6z2t21 t2 − v−5zt1t3 − 2v−7zt1t3 − v−7zt22 + v−8t4S3(568) = −v−2z4t−1t41 + 3v−3z3t−1t21 t2 − 2v−4z2t−1t1t3 − v−4z2t−1t22 + v−5zt−1t4 + z2t31 − 2v−1zt1t2 + v−2t3S3(568) = −v−1zt−1t4 + v−2t3S3(569) = −v−1zt2t4 + v−2t6S3(569) = v−4z4t41 t2−2v−5z3t31 t3−2v−5z3t21 t22+2v−6z2t21 t4+4v−6z2t1t2t3−2v−7zt1t5−v−7zt2t4−v−7zt23 +v−8t6S3(570) = −v−3z3t−2t31 + 2v−4z2t−2t1t2 − v−5zt−2t3 + v−2t1S3(570) = v−1zt−2t3 − z2t2−1t3 + v2t1S3(571) = −v−3zt−2t2 + v−1z3t−1t1 + v−1zt−1t1 − vz − vz−1 + v−1z + v−1z−1

S3(571) = −z2t−2t21 + v−1zt−2t2 + vz3t2−1t21 − z2t2−1t2 − vz3t−1t1 − vzt−1t1 − vz − vz−1 + v−1z + v−1z−1

S3(572) = vzt−2t3 + v3zt−1t2 + v4t1S3(572) = −v−5z3t−2t31 +2v−6z2t−2t1t2−v−7zt−2t3+v−4z4t2−1t31 −2v−5z3t2−1t1t2+v−6z2t2−1t3+v−4z2t−1t21 −v−5zt−1t2 + v−4t1S3(573) = −v−3z3t31 t3 + v−4z2t21 t4 + 2v−4z2t1t2t3 − v−5zt1t5 − v−5zt2t4 − v−5zt23 + v−6t6S3(573) = v−2z2t21 t4 − v−3zt1t5 − v−3zt2t4 + v−4t6S3(574) = v3zt1t3 + v2z2t4 + v2t4S3(574) = −v−5z5t41 − 2v−5z3t41 + 3v−6z4t21 t2 + 5v−6z2t21 t2 − 2v−7z3t1t3 − 3v−7zt1t3 − v−7z3t22 − v−7zt22 +v−8z2t4 + v−8t4S3(575) = −v−3zt−1t3 + v−2z2t2 + v−2t2S3(575) = vz3t−1t31 − 2z2t−1t1t2 + v−1zt−1t3 − vz3t21 − vzt21 + z2t2 + t2S3(576) = −vzt1t2 − z2t3 + v−2t3S3(576) = z2t31 − v−2z4t31 − v−2z2t31 − 2v−1zt1t2 + 2v−3z3t1t2 + v−3zt1t2 + v−2t3 − v−4z2t3S3(577) = z2t21 t2 + v−1z3t1t3 − v−1zt22 − v−2z2t4 + v−2t4S3(577) = −v−1z3t41 − v−2z4t21 t2 + 2v−2z2t21 t2 + v−3z3t1t3 − 2v−3zt1t3 + v−3z3t22 − v−4z2t4 + v−4t4S3(578) = v−1zt−1t2 + v2t1 − z2t1S3(578) = z2t−1t21 − v−1zt−1t2 − z2t1 + v−2t1S3(579) = v3zt−1t3 + v2z2t2 + v2t2S3(579) = −v−5z3t−1t31 + 2v−6z2t−1t1t2 − v−7zt−1t3 − v−3z3t21 − v−3zt21 + v−4z2t2 + v−4t2S3(580) = −v−3z3t1t3 − 2v−3zt1t3 + v−4z2t4 + v−4t4S3(580) = vz3t41 + z4t21 t2 − z2t21 t2 − v−1z3t1t3 − v−1z3t22 − v−1zt22 + v−2z2t4 + v−2t4S3(581) = v3zt22 + v2z2t4 + v2t4S3(581) = −v−5z5t41 − 2v−5z3t41 + 3v−6z4t21 t2 + 5v−6z2t21 t2 − 2v−7z3t1t3 − 2v−7zt1t3 − v−7z3t22 − 2v−7zt22 +v−8z2t4 + v−8t4S3(582) = vzt22 + v2t4S3(582) = −v−3z3t41 − v−5z3t41 + 2v−4z2t21 t2 + 3v−6z2t21 t2 − 2v−7zt1t3 − v−5zt22 − v−7zt22 + v−8t4S3(583) = v−1z3t−2t31 − 2v−2z2t−2t1t2 + v−3zt−2t3 + v2t1S3(583) = −v−3zt−2t3 + v−2z2t2−1t3 + v−2t1S3(584) = −v−1zt23 + v−2t6S3(584) = v−4z4t41 t2−2v−5z3t31 t3−2v−5z3t21 t22+3v−6z2t21 t4+2v−6z2t1t2t3−2v−7zt1t5+v−6z2t32−2v−7zt2t4+


v−8t6S3(585) = t6S3(585) = −v−5z5t61 + 5v−6z4t41 t2 − 4v−7z3t31 t3 − 6v−7z3t21 t22 + 3v−8z2t21 t4 + 6v−8z2t1t2t3 − 2v−9zt1t5 +v−8z2t32 − 2v−9zt2t4 − v−9zt23 + v−10t6S3(586) = v−1z3t−1t1 + v−1zt−1t1 − vz − 2vz−1 + 2v−1z + 3v−1z−1 − v−3z − v−3z−1

S3(586) = −vz3t−1t1 − vzt−1t1 + v3z + v3z−1 − 2vz − 3vz−1 + v−1z + 2v−1z−1

S3(587) = v5zt21 + v3zt21 + v4z2t2 + v4t2 + v2z2t2S3(587) = −v−3z3t21 − v−3zt21 − v−5z3t21 − 2v−5zt21 + v−4z2t2 + v−6z2t2 + v−6t2S3(588) = v5zt21 + v4z2t2 + v2z2t2 + v2t2S3(588) = −v−3z3t21 − v−3zt21 − v−5z3t21 − v−5zt21 + v−4z2t2 + v−4t2 + v−6z2t2S3(589) = v3z3t−1t1 + v3zt−1t1 + v7z−1 − 2v5z − 3v5z−1 + 2v3z + 2v3z−1

S3(589) = −v−3z3t−1t1 − v−3zt−1t1 − 2v−3z − 2v−3z−1 + 2v−5z + 3v−5z−1 − v−7z−1

S3(590) = v5zt−1t1 + v3zt−1t1 − v5z − v5z−1 + v3z−1 + vzS3(590) = −v−3zt−1t1 − v−5zt−1t1 − v−1z − v−3z−1 + v−5z + v−5z−1

S3(591) = v5zt−1t1 − v5z − v3z−1 + vz + vz−1

S3(591) = −v−5zt−1t1 − v−1z − v−1z−1 + v−3z−1 + v−5zS3(592) = v3zt−1t2 − v6t1 + v4z2t1 + 2v4t1 + v2z2t1S3(592) = v−4z2t−1t21 − v−5zt−1t2 + v−2z2t1 + v−4z2t1 + 2v−4t1 − v−6t1S3(593) = v5z3t21 + 2v5zt21 + v4z4t2 + 3v4z2t2 + v4t2S3(593) = −v−5z5t21 − 4v−5z3t21 − 3v−5zt21 + v−6z4t2 + 3v−6z2t2 + v−6t2S3(594) = v5z3t−1t1 + 2v5zt−1t1 − v5z3 − 3v5z − v5z−1 + v3z3 + 3v3z + v3z−1

S3(594) = −v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 3v−3z − v−3z−1 + v−5z3 + 3v−5z + v−5z−1

S3(595) = v5z3t−1t1 + v5zt−1t1 + v7z−1 − v5z3 − 3v5z − 3v5z−1 + v3z3 + 3v3z + 2v3z−1

S3(595) = −v−5z3t−1t1 − v−5zt−1t1 − v−3z3 − 3v−3z − 2v−3z−1 + v−5z3 + 3v−5z + 3v−5z−1 − v−7z−1

96 Knots in L(2, 1)

Knots in L(2, 1)

S3(11) = −v−1zt21 − v−1z−1 + v−3z−1

S3(21) = v3zt21 − v3z − v3z−1 + vz + vz−1

S3(21) = −vzt21 + vz − v−1z − v−1z−1 + v−3z−1

S3(22) = v−2z2t1 + 2v−2t1 − v−4t1S3(23) = t1S3(23) = v−2z2t31 + 2v−4z2t1 + 3v−4t1 − 2v−6t1S3(31) = −v4t1 + v2z2t1 + 2v2t1S3(31) = v2z2t31 − z4t1 − 2z2t1 + t1 + v−2z2t1S3(32) = −v−5z5t21 −4v−5z3t21 −3v−5zt21 +v−7zt21 −v−5z3−3v−5z−2v−5z−1+v−7z3+3v−7z+3v−7z−1−v−9z−1

S3(32) = −v−3z3t21 − 2v−3zt21 − v−3z − v−3z−1 + v−5z + v−5z−1

S3(33) = vzt21 − v3z−1 + vz−1

S3(33) = −v−1zt21 − v−3zt21 − v−3z−1 + v−5z−1

S3(34) = −v4z2t31 + v2z4t1 + 2v2z2t1 − z2t1 + t1S3(34) = v2t1 − z2t1 − t1 + v−2t1S3(35) = −v−3z3t21 − v−3zt21 − 2v−3z − 2v−3z−1 + 2v−5z + 3v−5z−1 − v−7z−1

S3(35) = −v−5z5t21 − 4v−5z3t21 − 4v−5zt21 + v−7zt21 − v−5z3 − 2v−5z − v−5z−1 + v−7z3 + 2v−7z + v−7z−1

S3(36) = −v−3z3t41 −3v−7z5t21 −9v−7z3t21 −6v−7zt21 +3v−9zt21 −3v−7z3−4v−7z−2v−7z−1+3v−9z3+4v−9z+3v−9z−1 − v−11z−1

S3(38) = v5zt21 − v5z − v3z−1 + vz + vz−1

S3(41) = −v3z3t41 + v−1z7t21 + 3v−1z5t21 + v−1z3t21 − 3v−1zt21 − v−3z3t21 + v−3zt21 + v−1z5 + 2v−1z3 + 3v−1z −v−3z5 − 2v−3z3 − 4v−3z − v−3z−1 + v−5z + v−5z−1

S3(42) = v−8z8t31 + 6v−8z6t31 + 10v−8z4t31 + 4v−8z2t31 − v−10z2t31 + 2v−10z8t1 + 14v−10z6t1 + 32v−10z4t1 +28v−10z2t1 + 8v−10t1 − 2v−12z6t1 − 11v−12z4t1 − 18v−12z2t1 − 9v−12t1 + 2v−14t1S3(42) = v−4z4t31 + 2v−4z2t31 + 2v−6z4t1 + 6v−6z2t1 + 4v−6t1 − 2v−8z2t1 − 3v−8t1S3(43) = v6z2t31 − v4z4t1 − 2v4z2t1 − v4t1 + 2v2z2t1 + 2v2t1S3(43) = v2z2t31 − z4t1 − 2z2t1 + 2v−2z2t1 + 2v−2t1 − v−4t1S3(44) = v3z3t21 + 2v3zt21 − vz3t21 − 2vzt21 − v3z3 − 2v3z + vz3 + 2vz − vz−1 + v−1z−1

S3(44) = −vz3t21 − 2vzt21 + v−1z3t21 + v−1zt21 + vz3 + 2vz − v−1z3 − 2v−1z − v−1z−1 + v−3z−1

S3(45) = v−6z6t31 +3v−6z4t31 +v−6z2t31 +3v−8z6t1+ 14v−8z4t1+ 18v−8z2t1+6v−8t1−3v−10z4t1− 10v−10z2t1−6v−10t1 + v−12t1S3(45) = v−6z6t31 +4v−6z4t31 +4v−6z2t31 +2v−8z6t1+9v−8z4t1+ 12v−8z2t1+5v−8t1−2v−10z4t1−6v−10z2t1−4v−10t1S3(46) = v5z3t21 + 2v5zt21 − v5z3 − 3v5z − v5z−1 + v3z3 + 3v3z + v3z−1

S3(46) = −v−1z3t21 − 2v−1zt21 + v−1z3 + 2v−1z − v−3z3 − 3v−3z − v−3z−1 + v−5z + v−5z−1

S3(47) = −v6t1 + v4z2t1 + v4t1 + v2z2t1 + v2t1S3(47) = z2t31 − v−2z4t1 − v−2z2t1 + 2v−2t1 + v−4z2t1 − v−4t1S3(48) = v−4z4t1 + 4v−4z2t1 + 3v−4t1 − v−6z2t1 − 2v−6t1S3(48) = −v2z2t31 + z4t1 + 2z2t1 + t1 − v−2z2t1S3(49) = −v−3zt21 − v−1z − v−1z−1 + v−3z + v−3z−1

S3(49) = v3zt21 − vz3t21 − vzt21 − vz − vz−1 + v−1z + v−1z−1

S3(410) = v5zt21 + v3zt21 − v5z − v5z−1 + v3z−1 + vzS3(410) = −vzt21 − v−1zt21 + vz − v−3z − v−3z−1 + v−5z−1

S3(411) = −z2t31 + z2t1 + 2t1 − v−2z2t1 − 2v−2t1 + v−4t1


S3(411) = v−2z2t1 + v−2t1 + v−4z2t1 + v−4t1 − v−6t1S3(412) = v3z3t21 + 2v3zt21 − vz3t21 − vzt21 − v3z3 − 2v3z − v3z−1 + vz3 + 2vz + vz−1

S3(412) = −vz3t21 − 2vzt21 + v−1z3t21 + v−1zt21 − v−3zt21 + vz3 + 2vz − v−1z3 − 2v−1z − v−3z−1 + v−5z−1

S3(413) = −v4z2t31 + v2z4t1 + 2v2z2t1 + v2t1 − 2z2t1 − t1 + v−2t1S3(414) = v−6z6t31 +3v−6z4t31 +2v−6z2t31 +3v−8z6t1+13v−8z4t1+16v−8z2t1+6v−8t1−3v−10z4t1−9v−10z2t1−6v−10t1 + v−12t1S3(415) = −v3z3t21 + vzt21 − v5z−1 + v3z3 + 2v3z + 2v3z−1 − vz3 − 3vz − 2vz−1 + v−1z + v−1z−1

S3(415) = v7z3t41 − v3z7t21 − 4v3z5t21 − 4v3z3t21 + v3zt21 + vz3t21 − 2vzt21 + 2v3z3 + v3z + v3z−1 − 3vz3 − 3vz −3vz−1 + v−1z3 + 2v−1z + 2v−1z−1

S3(416) = v−2z2t31 + v−4z2t31 + v−4z2t1 + v−4t1 + 2v−6z2t1 + 2v−6t1 − 2v−8t1S3(417) = v5z3t41 − vz7t21 − 3vz5t21 − vz3t21 + 2vzt21 + v−1z3t21 − 2v−1zt21 − vz5 − 2vz3 − 3vz − vz−1 + v−1z5 +2v−1z3 + 4v−1z + v−1z−1 − v−3zS3(417) = −v3zt21 + v3z − v−1z − v−1z−1 + v−3z−1

S3(418) = v−4z4t31 + v−4z2t31 + 3v−6z4t1 + 8v−6z2t1 + 4v−6t1 − 3v−8z2t1 − 3v−8t1S3(418) = v−8z8t31 + 6v−8z6t31 + 11v−8z4t31 + 6v−8z2t31 − v−10z2t31 + 2v−10z8t1 + 13v−10z6t1 + 28v−10z4t1 +24v−10z2t1 + 7v−10t1 − 2v−12z6t1 − 10v−12z4t1 − 15v−12z2t1 − 7v−12t1 + v−14t1S3(419) = −v3zt21 + vz3t21 + vzt21 − v3z−1 + 2vz + 2vz−1 − 2v−1z − 2v−1z−1 + v−3z−1

S3(419) = v5zt21 − v3z3t21 − v3zt21 + v5z−1 − 2v3z − 3v3z−1 + 2vz + 2vz−1

S3(420) = v5z3t21 + v5zt21 + v7z−1 − v5z3 − 3v5z − 3v5z−1 + v3z3 + 3v3z + 2v3z−1

S3(420) = −v5z3t41 + vz7t21 + 4vz5t21 + 4vz3t21 − 2vzt21 − v−1z3t21 + v−1zt21 − 2vz3 − vz−1 + 3v−1z3 + 3v−1z +2v−1z−1 − v−3z3 − 3v−3z − 2v−3z−1 + v−5z−1

S3(421) = v−6z6t31 +4v−6z4t31 +3v−6z2t31 +2v−8z6t1+10v−8z4t1+14v−8z2t1+5v−8t1−2v−10z4t1−7v−10z2t1−4v−10t1S3(423) = −v6z2t31 + v4z4t1 + 2v4z2t1 − v4t1 + 2v2t1S3(424) = v−4z4t51 +4v−10z8t31 + 18v−10z6t31 +25v−10z4t31 + 10v−10z2t31 −4v−12z2t31 +8v−12z8t1+39v−12z6t1+61v−12z4t1 + 40v−12z2t1 + 10v−12t1 − 8v−14z6t1 − 27v−14z4t1 − 28v−14z2t1 − 12v−14t1 + 3v−16t1S3(426) = −vzt21 + v−1z3t21 + v−1zt21 − vz−1 + v−1z + v−1z−1 − v−3zS3(51) = v8z2t31 − v6z4t1 − 3v6z2t1 − 2v6t1 + v4z4t1 + 5v4z2t1 + 3v4t1S3(51) = z4t31 + 2z2t31 − v−2z6t1 − 4v−2z4t1 − 3v−2z2t1 + v−2t1 + v−4z4t1 + 3v−4z2t1 + v−4t1 − v−6t1S3(52) = v5z3t21 + 2v5zt21 − v3z3t21 − v3zt21 − v5z3 − 2v5z + v3z3 + v3z − v3z−1 + vz + vz−1

S3(53) = −v6z2t31 + v4z4t1 + 2v4z2t1 − v2z2t1 + t1S3(54) = −v−7z7t21 −6v−7z5t21 −10v−7z3t21 −5v−7zt21 +v−9z3t21 +2v−9zt21 −v−7z5−5v−7z3−6v−7z−2v−7z−1+v−9z5 + 5v−9z3 + 7v−9z + 3v−9z−1 − v−11z − v−11z−1

S3(55) = v4z4t51 − v−2z10t31 −4v−2z8t31 − 3v−2z6t31 + 5v−2z4t31 +6v−2z2t31 + v−4z4t31 − 2v−4z2t31 − 2v−4z10t1 −10v−4z8t1−19v−4z6t1−21v−4z4t1−8v−4z2t1+2v−4t1+2v−6z8t1+7v−6z6t1+11v−6z4t1+10v−6z2t1−2v−8z2t1−v−8t1S3(56) = −v−11z11t41 −8v−11z9t41 − 21v−11z7t41 − 20v−11z5t41 − 5v−11z3t41 +v−13z3t41 − 3v−15z13t21 − 32v−15z11t21 −131v−15z9t21 − 260v−15z7t21 − 259v−15z5t21 − 120v−15z3t21 − 20v−15zt21 + 3v−17z9t21 + 23v−17z7t21 + 60v−17z5t21 +59v−17z3t21 +18v−17zt21 −3v−19zt21 −3v−15z11−27v−15z9−88v−15z7−131v−15z5−96v−15z3−35v−15z−5v−15z−1+3v−17z11 + 27v−17z9 + 89v−17z7 + 138v−17z5 + 115v−17z3 + 52v−17z + 10v−17z−1 − v−19z7 − 7v−19z5 − 19v−19z3 −17v−19z − 6v−19z−1 + v−21z−1

S3(56) = −v−5z5t41 −2v−5z3t41 −3v−9z7t21 −14v−9z5t21 −20v−9z3t21 −9v−9zt21 +3v−11z3t21 +5v−11zt21 −3v−9z5−9v−9z3 − 7v−9z − 2v−9z−1 + 3v−11z5 + 9v−11z3 + 8v−11z + 3v−11z−1 − v−13z − v−13z−1

S3(57) = −v7zt21 + 2v5z3t21 + 3v5zt21 − v5z3 − 2v5z − v5z−1 + v3z3 + 2v3z + v3z−1

S3(57) = −v5z3t41 + vz7t21 + 4vz5t21 + 4vz3t21 − vzt21 − 2v−1z3t21 − v−1zt21 − 2vz3 + 3v−1z3 + 2v−1z − v−3z3 −

98 Knots in L(2, 1)

2v−3z − v−3z−1 + v−5z−1

S3(58) = −v4z4t31 − 2v4z2t31 + v2z6t1 + 4v2z4t1 + 3v2z2t1 − v2t1 + z2t1 + 3t1 − v−2z2t1 − v−2t1S3(58) = z6t31 + 4z4t31 + 3z2t31 − v−2z4t31 − v−2z2t31 − v−2z8t1 − 6v−2z6t1 − 10v−2z4t1 − 3v−2z2t1 + 2v−2t1 +v−4z6t1 + 3v−4z4t1 + v−4z2t1 − v−4t1 + v−6z2t1S3(59) = −v−9z9t41 − 5v−9z7t41 − 6v−9z5t41 − v−9z3t41 − 4v−13z11t21 − 33v−13z9t21 − 98v−13z7t21 − 128v−13z5t21 −70v−13z3t21 − 12v−13zt21 + 4v−15z7t21 + 21v−15z5t21 + 29v−15z3t21 + 9v−15zt21 − v−17zt21 − 4v−13z9 − 27v−13z7 −62v−13z5−63v−13z3−30v−13z−5v−13z−1+4v−15z9+27v−15z7+64v−15z5+73v−15z3+44v−15z+ 10v−15z−1−2v−17z5 − 10v−17z3 − 14v−17z − 6v−17z−1 + v−19z−1

S3(59) = −v−7z7t41 − 4v−7z5t41 − 4v−7z3t41 − 3v−11z9t21 − 19v−11z7t21 − 42v−11z5t21 − 39v−11z3t21 − 13v−11zt21 +3v−13z5t21 + 10v−13z3t21 + 8v−13zt21 − 3v−11z7 − 14v−11z5 − 20v−11z3 − 10v−11z − 2v−11z−1 + 3v−13z7 + 14v−13z5 +21v−13z3 + 12v−13z + 3v−13z−1 − v−15z3 − 2v−15z − v−15z−1

S3(510) = v6z2t31 − v6t1 − v4z4t1 − v4z2t1 + v4t1 + 2v2z2t1 + v2t1S3(510) = v2z2t31 + z2t31 − z4t1 − 2z2t1 − v−2z4t1 + 2v−2t1 + v−4z2t1 − v−4t1S3(511) = −v7z3t41 +v3z7t21 +3v3z5t21 +2v3z3t21 −2vz3t21 +v3z5+v3z3+v3z−vz5−vz3−2vz−vz−1+v−1z+v−1z−1

S3(511) = −vz3t21 − vzt21 + v−1z3t21 + v−1zt21 − v−3zt21 + vz3 + vz − v−1z3 − 2v−1z − v−1z−1 + v−3z + v−3z−1

S3(512) = −v4z4t31 − v4z2t31 + v2z6t1 + 3v2z4t1 − 2v2t1 + z4t1 + 5z2t1 + 5t1 − 2v−2z2t1 − 2v−2t1S3(512) = z6t31 + 4z4t31 + 4z2t31 − v−2z4t31 − v−2z2t31 − v−2z8t1 − 6v−2z6t1 − 11v−2z4t1 − 6v−2z2t1 + v−2t1 +v−4z6t1 + 4v−4z4t1 + 5v−4z2t1 + v−4t1 − v−6t1S3(513) = v5zt21 − 2v3z3t21 − 2v3zt21 + vzt21 + v3z3 + v3z − vz3 − 2vz − vz−1 + v−1z + v−1z−1

S3(513) = v7z3t41 −v3z7t21 −4v3z5t21 −4v3z3t21 +2vz3t21 +2v3z3+v3z−3vz3−2vz−vz−1+v−1z3+v−1z+v−1z−1

S3(514) = −v2z6t31 − 4v2z4t31 − 3v2z2t31 + z4t31 + z2t31 + z8t1 + 6z6t1 + 10z4t1 + 4z2t1 − v−2z6t1 − 3v−2z4t1 +2v−2t1 − v−4z2t1 − v−4t1S3(514) = v2z4t31 + 2v2z2t31 − z6t1 − 4z4t1 − 3z2t1 + t1 − v−2t1 + 2v−4z2t1 + 2v−4t1 − v−6t1S3(515) = −v−9z9t41 −5v−9z7t41 −7v−9z5t41 −2v−9z3t41 −4v−13z11t21 −32v−13z9t21 −94v−13z7t21 − 125v−13z5t21 −74v−13z3t21 − 15v−13zt21 + 4v−15z7t21 + 20v−15z5t21 + 29v−15z3t21 + 11v−15zt21 − v−17zt21 − 4v−13z9 − 26v−13z7 −58v−13z5 − 54v−13z3 − 22v−13z − 3v−13z−1 +4v−15z9 + 26v−15z7 +60v−15z5 +62v−15z3 + 30v−15z + 5v−15z−1 −2v−17z5 − 8v−17z3 − 8v−17z − 2v−17z−1

S3(515) = −v−7z7t41 − 3v−7z5t41 − 2v−7z3t41 − 4v−11z9t21 − 24v−11z7t21 − 49v−11z5t21 − 40v−11z3t21 − 11v−11zt21 +4v−13z5t21 + 11v−13z3t21 + 7v−13zt21 − 4v−11z7 − 18v−11z5 − 26v−11z3 − 14v−11z − 3v−11z−1 + 4v−13z7 + 18v−13z5 +28v−13z3 + 18v−13z + 5v−13z−1 − 2v−15z3 − 4v−15z − 2v−15z−1

S3(516) = v4z4t31 + v4z2t1 + 2v4t1 − v2z6t1 − 3v2z4t1 − 4v2z2t1 − 3v2t1 + z4t1 + t1 + v−2t1S3(516) = −v6z4t31 − 2v6z2t31 + v4z4t31 + v4z2t31 + v4z6t1 + 4v4z4t1 + 3v4z2t1 − 2v4t1 − v2z6t1 − 3v2z4t1 +4v2t1 + z4t1 − t1S3(517) = −v−5z5t21 − 3v−5z3t21 − 2v−5zt21 − v−7z5t21 − 4v−7z3t21 − 2v−7zt21 + v−9zt21 − v−5z3 − v−5z − 2v−7z −2v−7z−1 + v−9z3 + 3v−9z + 3v−9z−1 − v−11z−1

S3(517) = −v−1z3t21 − 2v−1zt21 + v−3z5t21 + 3v−3z3t21 + 2v−3zt21 − v−5zt21 − v−1z − v−1z−1 + v−3z3 + 2v−3z +v−3z−1 − v−5z3 − v−5zS3(518) = −v5z3t21 − v5zt21 + v3z3t21 + 2v3zt21 + v5z3 + v5z − v3z3 − 2v3z − v3z−1 + vz + vz−1

S3(518) = −v3z3t41 + v−1z7t21 + 3v−1z5t21 + 2v−1z3t21 − v−1zt21 − 2v−3z3t21 − v−3zt21 + v−1z5 + v−1z3 + v−1z −v−3z5 − v−3z3 − 2v−3z − v−3z−1 + v−5z + v−5z−1

S3(519) = −v2z6t31 −4v2z4t31 −3v2z2t31 +z4t31 +2z2t31 +z8t1+6z6t1+10z4t1+3z2t1−2t1−v−2z6t1−3v−2z4t1+2v−2z2t1 + 6v−2t1 − v−4z2t1 − 3v−4t1S3(519) = v2z4t31 + 2v2z2t31 − z6t1 − 4z4t1 − 3z2t1 + t1 + v−4z2t1S3(520) = −v−7z7t41 − 3v−7z5t41 − v−7z3t41 − 4v−11z9t21 − 25v−11z7t21 − 52v−11z5t21 − 41v−11z3t21 − 9v−11zt21 +4v−13z5t21 + 12v−13z3t21 + 5v−13zt21 − 4v−11z7 − 19v−11z5 − 28v−11z3 − 17v−11z − 3v−11z−1 + 4v−13z7 + 19v−13z5 +


30v−13z3 + 22v−13z + 5v−13z−1 − 2v−15z3 − 5v−15z − 2v−15z−1

S3(520) = −v−9z9t41 −6v−9z7t41 −11v−9z5t41 −6v−9z3t41 −3v−13z11t21 −25v−13z9t21 −77v−13z7t21 −109v−13z5t21 −70v−13z3t21 − 16v−13zt21 + 3v−15z7t21 + 16v−15z5t21 + 25v−15z3t21 + 11v−15zt21 − 3v−13z9 − 20v−13z7 − 45v−13z5 −42v−13z3 − 17v−13z − 3v−13z−1 + 3v−15z9 + 20v−15z7 + 46v−15z5 + 46v−15z3 + 22v−15z + 5v−15z−1 − v−17z5 −4v−17z3 − 5v−17z − 2v−17z−1

S3(521) = v4z4t31 + 2v4z2t31 − v2z4t31 − v2z2t31 − v2z6t1 − 4v2z4t1 − 3v2z2t1 + 2v2t1 + z6t1 + 3z4t1 − 2t1 −v−2z4t1 − v−2z2t1 + v−4t1S3(521) = −v6z4t31 − v6z2t31 − v6z2t1 − 2v6t1 + v4z6t1 + 4v4z4t1 + 6v4z2t1 + 3v4t1 − v2z4t1 − v2z2t1S3(522) = −v3zt21 + v−1zt21 + v3z − vz−1 − v−1z + v−1z−1

S3(522) = v5z3t41 − vz7t21 − 3vz5t21 − vz3t21 + 2vzt21 + v−1z3t21 − 3v−1zt21 − vz5 − 2vz3 − 3vz + v−1z5 + 2v−1z3 +4v−1z − v−1z−1 − v−3z + v−3z−1

S3(523) = v4t1 − v2z2t1 − 2v2t1 + 2t1S3(524) = −v−3z3t41 −3v−7z5t21 −9v−7z3t21 −7v−7zt21 +3v−9zt21 −3v−7z3−3v−7z−v−7z−1+3v−9z3+3v−9z+v−9z−1

S3(525) = v7zt21 + v3z3t21 + v3zt21 − v7z − v5z − v5z−1 + 2v3z + v3z−1

S3(525) = −vz3t41 + v−3z7t21 + 3v−3z5t21 − 4v−3zt21 − 2v−5z3t21 + v−5zt21 + v−3z5 + 2v−3z3 + 3v−3z − v−5z5 −2v−5z3 − 5v−5z − v−5z−1 + 2v−7z + v−7z−1

S3(526) = −v6z4t31 − v6z2t31 + v4z6t1 + 3v4z4t1 + 2v4z2t1 − v4t1 − v2z4t1 + 2v2t1S3(526) = z2t1 + t1 − v−2z4t1 − 2v−2z2t1 + v−4z2t1S3(527) = −v6z4t31 − v6z2t31 + v4z6t1 + 3v4z4t1 + 2v4z2t1 − v4t1 − v2z4t1 + 2v2t1S3(527) = z2t1 + t1 − v−2z4t1 − 2v−2z2t1 + v−4z2t1S3(528) = −v−5z5t21 −3v−5z3t21 −2v−5zt21 −2v−5z3−5v−5z−2v−5z−1+2v−7z3+6v−7z+3v−7z−1−v−9z−v−9z−1

S3(528) = vz3t41 − v−3z7t21 − 3v−3z5t21 − 3v−3z3t21 − v−3zt21 + v−5z3t21 − v−5zt21 − v−3z5 − v−3z3 − 2v−3z −v−3z−1 + v−5z5 + v−5z3 + 2v−5z + v−5z−1

S3(529) = −v3z3t21 − v3zt21 + vzt21 + v3z3 + 2v3z − vz3 − 3vz − vz−1 + v−1z + v−1z−1

S3(529) = −v3zt21 + vz3t21 + vzt21 + v3z − vz3 − 2vz − vz−1 + v−1z3 + 2v−1z + v−1z−1 − v−3zS3(530) = v4t1 − v2z2t1 − v2t1 − z2t1 + v−2t1S3(530) = −v2z2t31 + v2t1 + z4t1 + z2t1 − t1 − v−2z2t1 + v−2t1S3(531) = v4z4t31 + v4z2t31 − v2z6t1 − 3v2z4t1 − 2v2z2t1 + z4t1 + z2t1 + t1S3(531) = v4z2t31 − v2z4t1 − 3v2z2t1 + z4t1 + 3z2t1 + t1 − v−2z2t1S3(532) = z2t1 + 2t1 − v−2z4t1 − 3v−2z2t1 − 2v−2t1 + v−4z2t1 + v−4t1S3(532) = z2t31 − v−2z4t1 − v−2z2t1 + v−2t1 + 2v−4z2t1 + v−4t1 − v−6t1S3(533) = v−1zt21 − v3z−1 + vz + vz−1 − v−1zS3(533) = −vzt21 + v−1z3t21 − v−3zt21 + v−1z − v−3z − v−3z−1 + v−5z−1

S3(534) = −2v−3z3t21 − 3v−3zt21 + v−5zt21 + v−3z3 + v−3z − v−3z−1 − v−5z3 − 2v−5z + v−5z−1 + v−7zS3(534) = −v−1z3t41 +v−5z7t21 +2v−5z5t21 −2v−5z3t21 −4v−5zt21 −v−7z3t21 +2v−7zt21 +v−5z5−2v−5z−2v−5z−1−v−7z5 + 2v−7z + 3v−7z−1 − v−9z−1

S3(535) = v7z3t21 + v5z3t21 + v5zt21 + v9z−1 − v7z3 − 2v7z − 2v7z−1 + v3z3 + 2v3z + v3z−1

S3(535) = −v3z3t41 + v−1z7t21 + 4v−1z5t21 + 3v−1z3t21 − 3v−1zt21 − v−3z3t21 + 2v−3zt21 − v−1z3 + v−1z − v−1z−1 +2v−3z3 + v−3z−1 − v−5z3 − v−5zS3(536) = −v2z2t1 − v2t1 + z4t1 + 3z2t1 + 3t1 − v−2z2t1 − v−2t1S3(536) = v−4z2t31 + v−4z4t1 + 3v−4z2t1 + 2v−4t1 − v−8t1S3(537) = vz3t21 + vzt21 − v−1zt21 − vz3 − 2vz − vz−1 + v−1z3 + 3v−1z + v−1z−1 − v−3zS3(537) = v5zt21 − v3z3t21 − v3zt21 − v5z + v3z3 + 2v3z − vz3 − 2vz − vz−1 + v−1z + v−1z−1

S3(538) = v7zt21 − v5z3t21 − v5zt21 + v7z−1 − 2v5z − 2v5z−1 + v3z + vz + vz−1

100 Knots in L(2, 1)

S3(539) = −v2z2t31 + z4t31 + z2t31 − z2t1 − t1 + 2v−2z4t1 + 5v−2z2t1 + 4v−2t1 − 2v−4z2t1 − 2v−4t1S3(540) = −v6z2t1 − 2v6t1 + v4z4t1 + 4v4z2t1 + 3v4t1S3(540) = z4t31 + 2z2t31 − v−2z6t1 − 4v−2z4t1 − 3v−2z2t1 + 2v−2t1 + v−4z4t1 + 2v−4z2t1 − v−4t1S3(541) = −v−7z7t21 − 6v−7z5t21 − 10v−7z3t21 − 4v−7zt21 + v−9z3t21 + 2v−9zt21 − v−7z5 − 5v−7z3 − 7v−7z −3v−7z−1 + v−9z5 + 5v−9z3 + 8v−9z + 5v−9z−1 − v−11z − 2v−11z−1

S3(542) = v2z2t31 + z2t31 − z4t1 − 2z2t1 + t1 − v−2z4t1 − v−2z2t1 + v−4z2t1S3(543) = v−1z3t41 − v−3z5t21 − 4v−3z3t21 − 3v−3zt21 + 2v−5z5t21 + 5v−5z3t21 + 4v−5zt21 − 2v−7zt21 − v−3z3 −3v−3z − 2v−3z−1 + 3v−5z3 + 4v−5z + 3v−5z−1 − 2v−7z3 − v−7z − v−7z−1

S3(543) = −v−3z3t21 − v−3zt21 − v−5z3t21 − 2v−5zt21 − v−3z − v−5z−1 + v−7z + v−7z−1

S3(544) = −v4z4t31 − 2v4z2t31 + v2z6t1 + 4v2z4t1 + 3v2z2t1 + t1 − v−2z2t1S3(544) = z6t31 + 4z4t31 + 3z2t31 − v−2z4t31 − v−2z2t31 + v−4z2t31 − v−2z8t1 − 6v−2z6t1 − 10v−2z4t1 − 3v−2z2t1 +2v−2t1 + v−4z6t1 + 3v−4z4t1 − 3v−4t1 + 3v−6z2t1 + 4v−6t1 − 2v−8t1S3(545) = v5z3t41 −vz7t21−3vz5t21−2vz3t21+2v−1z3t21−vz5−vz3−vz−vz−1+v−1z5+v−1z3+2v−1z+v−1z−1−v−3zS3(545) = v3z3t21 + v3zt21 − vz3t21 − 2vzt21 − v3z3 − v3z + vz3 + 2vz − v−1z − v−1z−1 + v−3z−1

S3(546) = −v6z4t31 − v6z2t1 − 2v6t1 + v4z6t1 + 3v4z4t1 + 4v4z2t1 + 3v4t1 − v2z4t1S3(546) = v4z4t31 + 2v4z2t31 − v2z4t31 − v2z2t31 − v2z6t1 − 4v2z4t1 − 3v2z2t1 + 2v2t1 + z6t1 + 3z4t1 − 3t1 −v−2z4t1 + 2v−2t1S3(547) = −v−9z9t41 −5v−9z7t41 −7v−9z5t41 −3v−9z3t41 −4v−13z11t21 −32v−13z9t21 −93v−13z7t21 − 121v−13z5t21 −69v−13z3t21 − 14v−13zt21 + 4v−15z7t21 + 20v−15z5t21 + 28v−15z3t21 + 11v−15zt21 − v−17zt21 − 4v−13z9 − 26v−13z7 −58v−13z5−57v−13z3−26v−13z−5v−13z−1+4v−15z9+26v−15z7+60v−15z5+66v−15z3+38v−15z+ 10v−15z−1−2v−17z5 − 9v−17z3 − 12v−17z − 6v−17z−1 + v−19z−1

S3(547) = −v−7z7t41 − 3v−7z5t41 − 2v−7z3t41 − 4v−11z9t21 − 24v−11z7t21 − 49v−11z5t21 − 40v−11z3t21 − 11v−11zt21 +4v−13z5t21 + 11v−13z3t21 + 6v−13zt21 − 4v−11z7 − 18v−11z5 − 26v−11z3 − 14v−11z − 2v−11z−1 + 4v−13z7 + 18v−13z5 +28v−13z3 + 18v−13z + 3v−13z−1 − 2v−15z3 − 4v−15z − v−15z−1

S3(548) = v6t1 − v4t1 − v2z4t1 − 3v2z2t1 − v2t1 + z2t1 + 2t1S3(548) = −v10z4t51 +v4z10t31 +5v4z8t31 +9v4z6t31 +4v4z4t31 −3v4z2t31 −v2z4t31 +3v2z2t31 +v2z10t1+2v2z8t1−2v2z6t1 + 2v2z2t1 − 2v2t1 − z8t1 + 3z4t1 + z2t1 + 5t1 − v−2z4t1 − v−2z2t1 − 2v−2t1S3(549) = −v−5z5t21 −3v−5z3t21 −v−5zt21 −v−7z5t21 −4v−7z3t21 −2v−7zt21 +v−9zt21 −v−5z3−2v−5z−v−5z−1−v−7z + v−9z3 + 3v−9z + 2v−9z−1 − v−11z−1

S3(549) = −v−1z3t21 − 2v−1zt21 + v−3z5t21 + 3v−3z3t21 + v−3zt21 − v−5zt21 − v−1z − v−1z−1 + v−3z3 + 3v−3z +2v−3z−1 − v−5z3 − 2v−5z − 2v−5z−1 + v−7z−1

S3(550) = v3z3t21 + 2v3zt21 − v5z − v5z−1 + v3z + v3z−1

S3(550) = −v−3z3t21 − 2v−3zt21 − v−5z3t21 − v−5zt21 − v−5z − v−5z−1 + v−7z + v−7z−1

S3(551) = v8z2t31 − v6z4t1 − 2v6z2t1 − v6t1 + 2v4z2t1 + v4t1 + v2z2t1 + v2t1S3(552) = −v2z4t31 − 2v2z2t31 + z6t1 + 4z4t1 + 4z2t1 + t1 − v−2z4t1 − 2v−2z2t1S3(552) = −v6z2t31 + v4z4t1 + 3v4z2t1 + v4t1 − v2z4t1 − 4v2z2t1 − 2v2t1 + z2t1 + 2t1S3(553) = v−1z3t21 + v−1zt21 − v−3zt21 − v−1z3 − 3v−1z − 2v−1z−1 + v−3z3 + 4v−3z + 3v−3z−1 − v−5z − v−5z−1

S3(553) = −v−7z7t21 −6v−7z5t21 −11v−7z3t21 −6v−7zt21 +v−9z3t21 +2v−9zt21 −v−7z5−4v−7z3−4v−7z−v−7z−1+v−9z5 + 4v−9z3 + 4v−9z + v−9z−1

S3(554) = −v6z4t31 − 2v6z2t31 + v4z6t1 + 4v4z4t1 + 4v4z2t1 − v4t1 − v2z4t1 − v2z2t1 + 2v2t1S3(555) = −v−1z3t21 − 2v−1zt21 + v−3z3t21 + v−3zt21 + v−1z3 + v−1z − v−1z−1 − v−3z3 − v−3z + v−3z−1

S3(555) = v5z3t21 +2v5zt21 −v3z3t21 −v3zt21 −vz3t21 −vzt21 −v5z3−2v5z+v3z3+2v3z−vz−vz−1+v−1z+v−1z−1

S3(556) = −v−5z5t21 −3v−5z3t21 −v−5zt21 −2v−5z3−6v−5z−3v−5z−1+2v−7z3+7v−7z+5v−7z−1−v−9z−2v−9z−1

S3(556) = vz3t41 − v−3z7t21 − 3v−3z5t21 − 3v−3z3t21 − 2v−3zt21 + v−5z3t21 − v−5zt21 − v−3z5 − v−3z3 − v−3z +v−5z5 + v−5z3 + v−5z − v−5z−1 + v−7z−1


S3(557) = v3zt21 + vzt21 − v5z−1 + v3z−1

S3(557) = −v−1zt21 − v−3zt21 − v−5zt21 − v−5z−1 + v−7z−1

S3(558) = −v6z2t31 − v4z2t31 + v4z4t1 + 2v4z2t1 + v2z4t1 + v2z2t1 − z2t1 + t1S3(558) = v2t1 − z2t1 − v−2z2t1 − v−2t1 + v−4t1S3(559) = −v−1z3t21 − v−1zt21 + v−3z5t21 + 3v−3z3t21 + 2v−3zt21 − v−5zt21 − 2v−1z − 2v−1z−1 + v−3z3 + 3v−3z +3v−3z−1 − v−5z3 − v−5z − v−5z−1

S3(559) = −v−5z5t21 − 3v−5z3t21 − 2v−5zt21 − v−7z5t21 − 4v−7z3t21 − 3v−7zt21 + v−9zt21 − v−5z3 − v−5z − v−7z −v−7z−1 + v−9z3 + 2v−9z + v−9z−1

S3(560) = −v4z2t31 + v4t1 + v2z4t1 + v2z2t1 − v2t1 − 2z2t1 + v−2t1S3(560) = −v4z2t31 − v2z2t31 + v2z4t1 + 2v2z2t1 + v2t1 + z4t1 − t1 − v−2z2t1 + v−2t1S3(561) = −v9z3t41 + v5z7t21 + 4v5z5t21 + 4v5z3t21 + v5zt21 − 2v3z3t21 + v3zt21 − 2v5z3 − 2v5z − v5z−1 + 3v3z3 +2v3z + v3z−1 − vz3

S3(561) = −v3zt21 + 2vz3t21 + vzt21 − 2v−1zt21 − vz3 + v−1z3 + 2v−1z − 2v−3z − v−3z−1 + v−5z−1

S3(562) = −v2z6t31 −4v2z4t31 −4v2z2t31 + z4t31 + z2t31 + z8t1+6z6t1+ 11z4t1+7z2t1+ t1−v−2z6t1−4v−2z4t1−4v−2z2t1S3(562) = v2z4t31 + v2z2t31 − z6t1 − 3z4t1 + 2t1 − v−2z4t1 − 4v−2z2t1 − 3v−2t1 + 3v−4z2t1 + 3v−4t1 − v−6t1S3(563) = v3z3t21 + v3zt21 − vz3t21 − vzt21 + v−1zt21 − v3z3 − v3z − v3z−1 + vz3 + 2vz + vz−1 − v−1zS3(563) = v5z3t41 − vz7t21 − 3vz5t21 − 2vz3t21 + 2v−1z3t21 − v−1zt21 − v−3zt21 − vz5 − vz3 − vz + v−1z5 + v−1z3 +2v−1z − v−3z − v−3z−1 + v−5z−1

S3(564) = −v6z4t31 − 2v6z2t31 + v4z4t31 + v4z2t31 + v4z6t1 + 4v4z4t1 + 3v4z2t1 − v4t1 − v2z6t1 − 3v2z4t1 −v2z2t1 + 2v2t1 + z4t1S3(564) = v4z4t31 −v2z2t31 +v4z2t1+2v4t1−v2z6t1−3v2z4t1−4v2z2t1−3v2t1+2z4t1+2z2t1+t1−v−2z2t1+v−2t1S3(565) = −v−7z7t41 − 2v−7z5t41 − v−7z3t41 − 5v−11z9t21 − 28v−11z7t21 − 52v−11z5t21 − 36v−11z3t21 − 8v−11zt21 +5v−13z5t21 + 11v−13z3t21 + 5v−13zt21 − 5v−11z7 − 22v−11z5 − 35v−11z3 − 22v−11z− 5v−11z−1 + 5v−13z7 + 22v−13z5 +39v−13z3 + 32v−13z + 10v−13z−1 − 4v−15z3 − 10v−15z − 6v−15z−1 + v−17z−1

S3(565) = −v−9z9t41 −5v−9z7t41 −8v−9z5t41 −4v−9z3t41 −4v−13z11t21 −31v−13z9t21 −89v−13z7t21 − 118v−13z5t21 −73v−13z3t21 − 17v−13zt21 + 4v−15z7t21 + 19v−15z5t21 + 28v−15z3t21 + 12v−15zt21 − v−17zt21 − 4v−13z9 − 25v−13z7 −54v−13z5 − 48v−13z3 − 18v−13z − 2v−13z−1 + 4v−15z9 + 25v−15z7 + 56v−15z5 + 55v−15z3 + 24v−15z + 3v−15z−1 −2v−17z5 − 7v−17z3 − 6v−17z − v−17z−1

S3(566) = −v−3z3t21 − v−5z3t21 − v−5zt21 − 2v−3z − v−3z−1 + 2v−7z + 2v−7z−1 − v−9z−1

S3(566) = v−1z3t41 − v−3z5t21 − 4v−3z3t21 − 4v−3zt21 + 2v−5z5t21 + 5v−5z3t21 + 3v−5zt21 − 2v−7zt21 − v−3z3 −2v−3z − v−3z−1 + 3v−5z3 + 4v−5z + 2v−5z−1 − 2v−7z3 − 2v−7z − 2v−7z−1 + v−9z−1

S3(567) = −v−3z3t41 −v−5z3t41 −2v−7z5t21 −5v−7z3t21 −3v−7zt21 −3v−9z5t21 −9v−9z3t21 −4v−9zt21 +3v−11zt21 −2v−7z3 − v−7z − v−9z3 − 3v−9z − 2v−9z−1 + 3v−11z3 + 4v−11z + 3v−11z−1 − v−13z−1

S3(568) = −v6z4t51 + z10t31 + 4z8t31 + 3z6t31 − 5z4t31 − 5z2t31 − v−2z4t31 + 3v−2z2t31 + 2v−2z10t1 + 10v−2z8t1 +19v−2z6t1+21v−2z4t1+10v−2z2t1+v−2t1−2v−4z8t1−7v−4z6t1−11v−4z4t1−8v−4z2t1+v−4t1+2v−6z2t1−v−6t1S3(568) = v4z2t31 − v2z4t1 − 2v2z2t1 + v2t1 − t1 + v−2t1S3(569) = −v−5z5t41 −v−5z3t41 −4v−9z7t21 −17v−9z5t21 −21v−9z3t21 −7v−9zt21 +4v−11z3t21 +4v−11zt21 −4v−9z5−11v−9z3 − 10v−9z − 3v−9z−1 + 4v−11z5 + 11v−11z3 + 12v−11z + 5v−11z−1 − 2v−13z − 2v−13z−1

S3(569) = −v−11z11t41 −8v−11z9t41 −22v−11z7t41 −24v−11z5t41 −8v−11z3t41 +v−13z3t41 −3v−15z13t21 −31v−15z11t21 −124v−15z9t21 − 244v−15z7t21 − 247v−15z5t21 − 121v−15z3t21 − 22v−15zt21 + 3v−17z9t21 + 22v−17z7t21 + 56v−17z5t21 +56v−17z3t21 +18v−17zt21 −2v−19zt21 −3v−15z11−26v−15z9−82v−15z7−118v−15z5−81v−15z3−26v−15z−3v−15z−1+3v−17z11 + 26v−17z9 + 83v−17z7 + 124v−17z5 + 95v−17z3 + 36v−17z + 5v−17z−1 − v−19z7 − 6v−19z5 − 14v−19z3 −10v−19z − 2v−19z−1

S3(570) = v4z2t31 − v2z4t31 − v2z2t31 + 2v2z2t1 + 3v2t1 − 3z4t1 − 8z2t1 − 5t1 + 3v−2z2t1 + 3v−2t1


S3(570) = −v8z2t31 + v6z4t1 + 2v6z2t1 − v6t1 + v4t1 + v2z2t1 + v2t1S3(571) = v9z3t41 − v5z7t21 − 4v5z5t21 − 4v5z3t21 + 2v5zt21 − 3v3zt21 + 2v5z3 + v5z−1 − 2v3z3 − v3z − 2v3z−1 +v−1z + v−1z−1

S3(572) = −v8t1 + v4z4t1 + 4v4z2t1 + 2v4t1S3(572) = v8z4t51 − v2z10t31 − 5v2z8t31 − 9v2z6t31 − 4v2z4t31 + 4v2z2t31 + z4t31 − 2z2t31 − z10t1 − 2z8t1 + 2z6t1 −z4t1 − 4z2t1 + 3t1 + v−2z8t1 − 4v−2z4t1 − 2v−2z2t1 − 4v−2t1 + v−4z4t1 + 2v−4z2t1 + 2v−4t1S3(573) = −v−9z9t41 −6v−9z7t41 −10v−9z5t41 −4v−9z3t41 −3v−13z11t21 −26v−13z9t21 −82v−13z7t21 −116v−13z5t21 −71v−13z3t21 − 14v−13zt21 + 3v−15z7t21 + 17v−15z5t21 + 26v−15z3t21 + 9v−15zt21 − 3v−13z9 − 21v−13z7 − 49v−13z5 −48v−13z3 − 21v−13z − 3v−13z−1 + 3v−15z9 + 21v−15z7 + 50v−15z5 + 53v−15z3 + 28v−15z + 5v−15z−1 − v−17z5 −5v−17z3 − 7v−17z − 2v−17z−1

S3(573) = −v−7z7t41 − 4v−7z5t41 − 3v−7z3t41 − 3v−11z9t21 − 20v−11z7t21 − 45v−11z5t21 − 40v−11z3t21 − 11v−11zt21 +3v−13z5t21 + 11v−13z3t21 + 7v−13zt21 − 3v−11z7 − 15v−11z5 − 22v−11z3 − 13v−11z − 3v−11z−1 + 3v−13z7 + 15v−13z5 +23v−13z3 + 16v−13z + 5v−13z−1 − v−15z3 − 3v−15z − 2v−15z−1

S3(575) = v7z3t41 − v3z7t21 − 3v3z5t21 − v3z3t21 + 3v3zt21 − 3vzt21 − v3z5 − 2v3z3 − 3v3z + vz5 + 2vz3 + 3vz −vz−1 + v−1z−1

S3(576) = z2t31 − v−2z4t31 − v−2z2t31 + 2v−2z2t1 + 3v−2t1 − 2v−4z4t1 − 4v−4z2t1 − 3v−4t1 + 2v−6z2t1 + v−6t1S3(577) = −v−1z3t41 +v−5z7t21 +2v−5z5t21 −3v−5z3t21 −5v−5zt21 −v−7z3t21 +2v−7zt21 +v−5z5+v−5z3−v−5z−1−v−7z5 − v−7z3 − v−7z + v−7z−1 + v−9zS3(579) = v7zt21 − v7z − v3z−1 + vz + vz−1

S3(579) = −vz3t41 + v−3z7t21 + 3v−3z5t21 − 4v−3zt21 − v−5z3t21 + 2v−5zt21 + v−3z5 + 2v−3z3 + 2v−3z − v−3z−1 −v−5z5 − 2v−5z3 − 3v−5z + v−5z−1 + v−7zS3(580) = −v−5z5t21 − 4v−5z3t21 − 3v−5zt21 − v−5z3 − 3v−5z − v−5z−1 + v−7z3 + 3v−7z + v−7z−1

S3(580) = vz3t41 − v−3z7t21 − 3v−3z5t21 − 2v−3z3t21 + v−5z3t21 − v−5zt21 − v−3z5 − 2v−3z3 − 4v−3z − 2v−3z−1 +v−5z5 + 2v−5z3 + 5v−5z + 3v−5z−1 − v−7z − v−7z−1

S3(581) = −v−5z5t41 −2v−5z3t41 −3v−9z7t21 −14v−9z5t21 −20v−9z3t21 −8v−9zt21 +3v−11z3t21 +5v−11zt21 −3v−9z5−9v−9z3 − 8v−9z − 3v−9z−1 + 3v−11z5 + 9v−11z3 + 9v−11z + 5v−11z−1 − v−13z − 2v−13z−1

S3(582) = −vzt21 + v−1z3t21 − vz−1 + 2v−1z + 2v−1z−1 − 2v−3z − 2v−3z−1 + v−5z−1

S3(582) = −v−3z3t41 −v−5z3t41 −2v−7z5t21 −5v−7z3t21 −2v−7zt21 −3v−9z5t21 −9v−9z3t21 −4v−9zt21 +3v−11zt21 −2v−7z3 − 2v−7z − v−7z−1 − v−9z3 − 2v−9z + 3v−11z3 + 4v−11z + 2v−11z−1 − v−13z−1

S3(583) = −v6z2t31 + v4z4t31 + v4z2t31 − 2v4z2t1 − 3v4t1 + 3v2z4t1 + 8v2z2t1 + 6v2t1 − 3z2t1 − 2t1S3(583) = v6z2t31 − v4z4t1 − 2v4z2t1 + v4t1 − v2t1 − z2t1 + v−2t1S3(584) = −v−5z5t41 −v−5z3t41 −4v−9z7t21 −17v−9z5t21 −21v−9z3t21 −7v−9zt21 +4v−11z3t21 +3v−11zt21 −4v−9z5−11v−9z3 − 10v−9z − 2v−9z−1 + 4v−11z5 + 11v−11z3 + 12v−11z + 3v−11z−1 − 2v−13z − v−13z−1

S3(584) = −v−11z11t41 −8v−11z9t41 −22v−11z7t41 −24v−11z5t41 −9v−11z3t41 +v−13z3t41 −3v−15z13t21 −31v−15z11t21 −124v−15z9t21 − 243v−15z7t21 − 243v−15z5t21 − 116v−15z3t21 − 21v−15zt21 + 3v−17z9t21 + 22v−17z7t21 + 56v−17z5t21 +55v−17z3t21 +18v−17zt21 −2v−19zt21 −3v−15z11−26v−15z9−82v−15z7−118v−15z5−84v−15z3−30v−15z−5v−15z−1+3v−17z11 + 26v−17z9 + 83v−17z7 + 124v−17z5 + 99v−17z3 + 44v−17z + 10v−17z−1 − v−19z7 − 6v−19z5 − 15v−19z3 −14v−19z − 6v−19z−1 + v−21z−1

S3(585) = −v−5z5t61 −5v−13z11t41 −30v−13z9t41 −64v−13z7t41 −55v−13z5t41 −15v−13z3t41 +5v−15z3t41 −15v−17z13t21 −121v−17z11t21 −378v−17z9t21 −587v−17z7t21 −475v−17z5t21 − 190v−17z3t21 −30v−17zt21 + 15v−19z9t21 +76v−19z7t21 +140v−19z5t21 +105v−19z3t21 +30v−19zt21 −6v−21zt21 −15v−17z11−96v−17z9−222v−17z7−238v−17z5−136v−17z3−40v−17z−5v−17z−1+15v−19z11+96v−19z9+227v−19z7+254v−19z5+167v−19z3+60v−19z+10v−19z−1−5v−21z7−16v−21z5 − 31v−21z3 − 20v−21z − 6v−21z−1 + v−23z−1

S3(586) = vz3t21 + vzt21 − vz3 − 2vz − 2vz−1 + v−1z3 + 3v−1z + 3v−1z−1 − v−3z − v−3z−1

S3(586) = −v3z3t21 − v3zt21 + v3z3 + 2v3z + v3z−1 − vz3 − 3vz − 3vz−1 + v−1z + 2v−1z−1


S3(588) = −v−3z3t21 − v−3zt21 − v−5z3t21 − v−5zt21 − v−3z − v−3z−1 + v−5z−1 + v−7zS3(589) = −v−1z3t21 − v−1zt21 + v−1z3 + v−1z − v−3z3 − 3v−3z − 2v−3z−1 + 2v−5z + 3v−5z−1 − v−7z−1

S3(590) = v7zt21 + v5zt21 − v7z − v5z − v5z−1 + v3z + v3z−1 + vzS3(592) = −2v6t1 + 2v4z2t1 + 3v4t1 + v2z2t1S3(594) = v7z3t21 + 2v7zt21 − v7z3 − 2v7z − v5z − v5z−1 + v3z3 + 3v3z + v3z−1

S3(595) = v7z3t21 + v7zt21 − v7z3 − v7z + v7z−1 − 2v5z − 3v5z−1 + v3z3 + 3v3z + 2v3z−1

Knots in L(3, 1)

S3(11) = t−1S3(11) = −v−1zt2−1 + v−2t1S3(22) = −v−3zt−1t1 − v−1z − v−1z−1 + v−3z + v−3z−1

S3(23) = v−2z2t3−1 − 2v−5zt−1t1 − 2v−3z − v−3z−1 + 2v−5z + v−5z−1

S3(31) = v3zt2−1 + v2z2t1 + v2t1S3(31) = v−2z2t2−1t1 − v−2z2t−1 + v−2t−1 − v−1zt21S3(32) = v−6z2t2−1t1 + v−4z4t−1 + 4v−4z2t−1 + 2v−4t−1 − 2v−6z2t−1 − v−6t−1 − v−5zt21S3(32) = v−2z2t1 + 2v−2t1 − v−4t1S3(33) = v2t−1 + vzt21S3(33) = −v−1zt2−1 − v−3zt2−1 + v−4t1S3(34) = vzt2−1 − z2t−1t21 + v2t1 + z2t1S3(34) = −z2t−1 + v−2t−1 − vzt21S3(35) = −v−3zt2−1 + v−2z2t1 + v−2t1S3(35) = v−6z2t2−1t1 + v−4z4t−1 + 4v−4z2t−1 + 3v−4t−1 − 2v−6z2t−1 − 2v−6t−1S3(36) = −v−3z3t4−1 + 3v−8z2t2−1t1 + 3v−6z4t−1 + 9v−6z2t−1 + 3v−6t−1 − 6v−8z2t−1 − 2v−8t−1 − v−7zt21S3(37) = −v−3z3t2−1 − 2v−3zt2−1 + v−4z2t1 + v−4t1S3(41) = −v4t−1 + v2z2t−1 + 2v2t−1S3(41) = −v−3z3t3−1t1 + 2v−3z3t2−1 − v−3zt2−1 + 2v−4z2t−1t21 + 2v−2z2t1 + v−2t1 − 3v−4z2t1S3(42) = −v−9z3t3−1t1−v−7z7t2−1−6v−7z5t2−1− 10v−7z3t2−1−3v−7zt2−1+3v−9z3t2−1+v−9zt2−1+2v−10z2t−1t21 +v−8z6t1 + 5v−8z4t1 + 9v−8z2t1 + 3v−8t1 − 4v−10z2t1 − 2v−10t1S3(44) = −z2t2−1t1 − v2z2t−1 − v2t−1 + 2z2t−1 + 2t−1 − vz3t21 − vzt21S3(44) = v−1z3t2−1 + v−2z2t−1t21 + z2t1 + t1 − 2v−2z2t1S3(45) = v−8z2t2−1t1 + 2v−6z4t−1 + 6v−6z2t−1 + 3v−6t−1 − 2v−8z2t−1 − 2v−8t−1 − v−5z5t21 − 3v−5z3t21 − v−5zt21S3(45) = v−6z4t−1 + 2v−6z2t−1 + v−6t−1 − v−5z5t21 − 4v−5z3t21 − 4v−5zt21 + v−7zt21S3(47) = v5zt2−1 + v4z2t1 + v2z2t1 + v2t1S3(47) = v−4z2t2−1t1 + v−2z2t−1 + v−2t−1 − v−4z2t−1 − v−3zt21S3(48) = −v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 3v−3z − v−3z−1 + v−5z3 + 3v−5z + v−5z−1

S3(48) = −v−1z3t−1t1 − v2z2t31 − vz3 − vz − vz−1 + v−1z3 + v−1z + v−1z−1

S3(49) = z2t−1 + t−1 + v3zt21 − vz3t21 − vzt21S3(411) = −v−1zt−1t1 + v−3zt−1t1 − z2t31 − vz − vz−1 + 2v−1z + v−1z−1 − v−3zS3(411) = −v−3zt−1t1 − v−5zt−1t1 − v−1z − v−3z−1 + v−5z + v−5z−1

S3(412) = −z2t2−1t1 − v2z2t−1 + 2z2t−1 + t−1 − vz3t21S3(412) = v−1z3t2−1 − v−3zt2−1 + v−2z2t−1t21 + z2t1 + t1 − 2v−2z2t1 − v−2t1 + v−4t1S3(414) = v−8z2t2−1t1 +2v−6z4t−1 + 5v−6z2t−1 +2v−6t−1 −2v−8z2t−1 −v−8t−1 −v−5z5t21 − 3v−5z3t21 −2v−5zt21


S3(415) = −vz3t−1t1 − vzt−1t1 + v−1zt−1t1 + v3z − vz − vz−1 + v−1z−1

S3(415) = −v2z2t3−1 + v−1z3t2−1t21 − vzt−1t1 − 3v−1z3t−1t1 + v−1zt−1t1 − v2z2t31 − 2vz3 − vz − vz−1 + 2v−1z3 +v−1z + v−1z−1

S3(416) = v−1zt−1t1 − v3z−1 + vz + vz−1 − v−1zS3(416) = v−2z2t3−1 + v−4z2t3−1 − v−5zt−1t1 − 2v−7zt−1t1 − v−3z − v−5z − v−5z−1 + 2v−7z + v−7z−1

S3(417) = −2v−2z2t2−1t1 + v−1z3t−1t31 − 2z2t−1 + 3v−2z2t−1 + v−2t−1 − vzt21 − 2v−1z3t21S3(417) = v2t−1 − z2t−1 − t−1 + v−2t−1S3(418) = 2v−4z2t−1 + 2v−4t−1 − v−6t−1 − v−3z3t21 − v−3zt21S3(418) = −v−9z3t3−1t1−v−7z7t2−1−6v−7z5t2−1− 11v−7z3t2−1−5v−7zt2−1+3v−9z3t2−1+2v−9zt2−1+v−10z2t−1t21 +v−8z6t1 + 4v−8z4t1 + 6v−8z2t1 + 2v−8t1 − 2v−10z2t1 − v−10t1S3(419) = v−1z3t−1t1 − v−1zt−1t1 + v2z2t31 + vz3 + vz − v−1z3 − v−1z − v−1z−1 + v−3z−1

S3(419) = −vz3t−1t1 + vzt−1t1 − v4z2t31 − v3z3 − v3z − v3z−1 + vz3 + vz + vz−1

S3(420) = z2t3−1−v−3z3t2−1t21 +3v−3z3t−1t1−2v−3zt−1t1+z2t31 +2v−1z3+v−1z−2v−3z3−v−3z−v−3z−1+v−5z−1

S3(421) = v−6z4t−1 + 3v−6z2t−1 + 2v−6t−1 − v−8t−1 − v−5z5t21 − 4v−5z3t21 − 3v−5zt21 + v−7zt21S3(422) = v−4z4t3−1+2v−4z2t3−1−2v−7z3t−1t1−3v−7zt−1t1−2v−5z3−4v−5z−v−5z−1+2v−7z3+4v−7z+v−7z−1

S3(423) = v3zt2−1 − v2z2t−1t21 + 2v2z2t1 + v2t1S3(424) = v−4z4t5−1−4v−11z3t3−1t1−4v−9z7t2−1−18v−9z5t2−1−25v−9z3t2−1−6v−9zt2−1+12v−11z3t2−1+3v−11zt2−1+3v−12z2t−1t21 + 4v−10z6t1 + 9v−10z4t1 + 12v−10z2t1 + 3v−10t1 − 6v−12z2t1 − 2v−12t1S3(426) = t−1 − v−2z2t−1 − vzt21 + v−1z3t21 + v−1zt21S3(427) = v−3z3t−1t1 − v−3zt−1t1 + z2t31 + v−1z3 − v−1z−1 − v−3z3 + v−3z−1

S3(51) = v5z3t2−1 + v5zt2−1 + v4z2t−1t21 − v6t1 + v4z4t1 + 2v4z2t1 + 2v4t1S3(51) = v−4z4t2−1t1 + 2v−4z2t2−1t1 − v−4z4t−1 + 2v−4t−1 − v−6t−1 − v−3z3t21 − v−3zt21S3(52) = −v2z2t2−1t1 − v4z2t−1 − v4t−1 + 3v2z2t−1 + 2v2t−1 − v3z3t21S3(54) = v−8z4t2−1t1 + 2v−8z2t2−1t1 + v−6z6t−1 + 6v−6z4t−1 + 9v−6z2t−1 + 3v−6t−1 − 2v−8z4t−1 − 5v−8z2t−1 −2v−8t−1 − v−7z3t21 − v−7zt21S3(55) = v−4z4t4−1t1 − 3v−4z4t3−1 + v−4z2t3−1 − 3v−7z3t2−1t21 − 3v−5z5t−1t1 − 9v−5z3t−1t1 − 3v−5zt−1t1 +10v−7z3t−1t1 + v−4z2t31 + 4v−5z3 − v−5z − v−5z−1 − 4v−7z3 + v−7z + v−7z−1

S3(56) = v−12z4t4−1t1 + v−10z10t3−1 + 8v−10z8t3−1 + 21v−10z6t3−1 + 20v−10z4t3−1 + 4v−10z2t3−1 − 4v−12z4t3−1 −v−12z2t3−1−3v−15z3t2−1t21−2v−13z9t−1t1−15v−13z7t−1t1−39v−13z5t−1t1−40v−13z3t−1t1−9v−13zt−1t1+12v−15z3t−1t1+4v−15zt−1t1 + v−12z2t31 − 2v−11z9 − 16v−11z7 − 43v−11z5 − 46v−11z3 − 17v−11z − 2v−11z−1 + 2v−13z9 + 16v−13z7 +43v−13z5 + 52v−13z3 + 21v−13z + 3v−13z−1 − 6v−15z3 − 4v−15z − v−15z−1

S3(57) = v6z2t3−1 + 2v3z3t−1t1 + 2v3zt−1t1 + v5z3 − v5z−1 − v3z3 + v3z−1

S3(57) = −v−3z3t2−1t21 + 2v−3z3t−1t1 − 2v−3zt−1t1 + z2t31 + v−1z3 − v−3z3 − v−3z−1 + v−5z−1

S3(58) = vz3t−1t1 + 2vzt−1t1 − v−1z3t−1t1 − 2v−1zt−1t1 − vz3 − 2vz − vz−1 + v−1z3 + 2v−1z + v−1z−1

S3(58) = −v−2z4t3−1 − v−5z3t2−1t21 − v−3z5t−1t1 − 4v−3z3t−1t1 − 2v−3zt−1t1 + 5v−5z3t−1t1 + v−2z2t31 + 3v−3z3 −v−3z−1 − 3v−5z3 + v−5z−1

S3(59) = v−8z8t3−1+5v−8z6t3−1+6v−8z4t3−1+v−8z2t3−1−v−13z3t2−1t21−3v−11z7t−1t1−16v−11z5t−1t1−22v−11z3t−1t1−6v−11zt−1t1+4v−13z3t−1t1+2v−13zt−1t1+v−10z2t31 −3v−9z7−17v−9z5−27v−9z3−13v−9z−2v−9z−1+3v−11z7+17v−11z5 + 29v−11z3 + 15v−11z + 3v−11z−1 − 2v−13z3 − 2v−13z − v−13z−1

S3(59) = −2v−9z5t−1t1 − 6v−9z3t−1t1 − 4v−9zt−1t1 + v−6z6t31 + 4v−6z4t31 + 4v−6z2t31 − 2v−7z5 − 7v−7z3 −6v−7z − v−7z−1 + 2v−9z5 + 7v−9z3 + 6v−9z + v−9z−1

S3(510) = v5zt2−1 + v2z2t−1t21 + v4z2t1 + v2t1S3(510) = v−2z2t2−1t1 + v−4z2t2−1t1 + v−2t−1 − v−4z2t−1 − v−3zt21S3(511) = z2t2−1t1 − vz3t−1t31 + v2z2t−1 − z2t−1 + t−1 + v3zt21 + vz3t21 − vzt21S3(511) = v−1z3t2−1 − v−3zt2−1 + v−2z2t−1t21 + z2t1 − v−2z2t1 + v−2t1


S3(512) = −v2z2t3−1+vz3t−1t1+vzt−1t1−2v−1z3t−1t1−v−1zt−1t1−2vz3−2vz−vz−1+2v−1z3+2v−1z+v−1z−1

S3(512) = −v−2z4t3−1−v−5z3t2−1t21 −v−3z5t−1t1−4v−3z3t−1t1−3v−3zt−1t1+4v−5z3t−1t1+v−5zt−1t1+2v−3z3−v−3z−1 − 2v−5z3 + v−5z−1

S3(513) = −v4z2t3−1 − 2vz3t−1t1 − vzt−1t1 + v−1zt−1t1 − v3z3 + vz3 − vz−1 + v−1z−1

S3(513) = v−1z3t2−1t21 − vzt−1t1 − 2v−1z3t−1t1 + v−1zt−1t1 − v2z2t31 − vz3 − vz−1 + v−1z3 + v−1z−1

S3(514) = −z2t3−1 + v−3z3t2−1t21 + v−1z5t−1t1 + 4v−1z3t−1t1 + v−1zt−1t1 − 5v−3z3t−1t1 − v−3zt−1t1 + z4t31 − vz −vz−1 − 3v−1z3 + v−1z−1 + 3v−3z3 + v−3zS3(514) = −v−1z3t−1t1 − 2v−1zt−1t1 + v−3z3t−1t1 + v−3zt−1t1 − v−5zt−1t1 + v−1z3 + v−1z − v−3z3 − 2v−3z −v−3z−1 + v−5z + v−5z−1

S3(515) = v−8z8t3−1 + 5v−8z6t3−1 + 7v−8z4t3−1 + 2v−8z2t3−1 − v−13z3t2−1t21 − 3v−11z7t−1t1 − 15v−11z5t−1t1 −21v−11z3t−1t1 − 7v−11zt−1t1 + 4v−13z3t−1t1 + 3v−13zt−1t1 − 3v−9z7 − 16v−9z5 − 25v−9z3 − 13v−9z − 2v−9z−1 +3v−11z7 + 16v−11z5 + 27v−11z3 + 16v−11z + 3v−11z−1 − 2v−13z3 − 3v−13z − v−13z−1

S3(515) = −3v−9z5t−1t1 − 8v−9z3t−1t1 − 5v−9zt−1t1 + v−11zt−1t1 + v−6z6t31 + 3v−6z4t31 + 2v−6z2t31 − 3v−7z5 −10v−7z3 − 8v−7z − v−7z−1 + 3v−9z5 + 10v−9z3 + 9v−9z + v−9z−1 − v−11zS3(516) = z4t2−1t1 + z2t2−1t1 + v2t−1 − z4t−1 − 2z2t−1 − t−1 + v−2t−1S3(516) = vz3t3−1t1 + v3z3t2−1 + v3zt2−1 − 3vz3t2−1 − vzt2−1 + z4t−1t21 − z2t−1t21 + v2z4t1 − z4t1 + 2z2t1 + t1S3(517) = v−6z2t2−1t1+v−8z2t2−1t1+v−4z4t−1+3v−4z2t−1+v−4t−1+v−6z4t−1+2v−6z2t−1+v−6t−1−2v−8z2t−1−v−8t−1 − v−7zt21S3(517) = −v−4z2t−1t21 + z2t1 + 2t1 − v−2z4t1 − 3v−2z2t1 − 2v−2t1 + 2v−4z2t1 + v−4t1S3(518) = v2z2t2−1t1 + v4z2t−1 − v2z2t−1 + v2t−1 + v3z3t21 + v3zt21S3(518) = −v−3z3t3−1t1 + v−3z3t2−1 − 2v−3zt2−1 + v−4z2t−1t21 + v−2z2t1 − v−4z2t1 + v−4t1S3(519) = −z2t3−1 + v−3z3t2−1t21 + v−1z5t−1t1 + 4v−1z3t−1t1 + 2v−1zt−1t1 − 5v−3z3t−1t1 − 3v−3zt−1t1 + z4t31 +z2t31 − 3v−1z3 − 3v−1z − v−1z−1 + 3v−3z3 + 3v−3z + v−3z−1

S3(519) = −v−1z3t−1t1 − 2v−1zt−1t1 + v−3z3t−1t1 + v−3zt−1t1 + v−1z3 + v−1z − v−1z−1 − v−3z3 − v−3z + v−3z−1

S3(520) = −3v−9z5t−1t1 − 9v−9z3t−1t1 − 4v−9zt−1t1 + v−11zt−1t1 + v−6z6t31 + 3v−6z4t31 + v−6z2t31 − 3v−7z5 −11v−7z3 − 9v−7z − 2v−7z−1 + 3v−9z5 + 11v−9z3 + 10v−9z + 3v−9z−1 − v−11z − v−11z−1

S3(520) = v−8z8t3−1 + 6v−8z6t3−1 + 11v−8z4t3−1 + 6v−8z2t3−1 − v−10z2t3−1 − 2v−11z7t−1t1 − 10v−11z5t−1t1 −14v−11z3t−1t1 −6v−11zt−1t1 +v−13zt−1t1 −2v−9z7 − 11v−9z5 − 18v−9z3 − 10v−9z−v−9z−1 +2v−11z7 + 11v−11z5 +18v−11z3 + 11v−11z + v−11z−1 − v−13zS3(521) = −v−2z4t2−1t1 + v−2z2t2−1t1 − v−1z3t−1t31 − z4t−1 + t−1 + v−2z4t−1 − 3v−2z2t−1 − v−2t−1 + v−4t−1 −vz3t21 − vzt21 + 3v−1z3t21S3(521) = v3zt2−1 − v2z4t−1t21 − 2v2z2t−1t21 + v2z4t1 + 3v2z2t1 + v2t1S3(522) = v2t−1 − z2t−1 + v−1zt21S3(522) = v−1z3t3−1t1 − vzt2−1 − 2v−1z3t2−1 − v−1zt2−1 − 2v−2z2t−1t21 − 2z2t1 − t1 + 3v−2z2t1 + 2v−2t1S3(523) = −v−2z2t2−1t1 + 2t−1 + v−2z2t−1 − v−2t−1 + v−1zt21S3(523) = −v3zt2−1 − v2z2t1 − v2t1 + 2t1S3(524) = −v−3z3t4−1 + 3v−8z2t2−1t1 + 3v−6z4t−1 + 9v−6z2t−1 + 4v−6t−1 − 6v−8z2t−1 − 3v−8t−1S3(525) = −v6t−1 + 2v4z2t−1 + 2v4t−1 + v3z3t21 + v3zt21S3(525) = −v−5z3t3−1t1−v−3z3t2−1−v−3zt2−1+v−5z3t2−1−v−5zt2−1+2v−6z2t−1t21 +2v−4z2t1+v−4t1−2v−6z2t1S3(526) = v3z3t2−1 + v3zt2−1 − v2z4t−1t21 − v2z2t−1t21 + v4z2t1 + v2z4t1 + v2z2t1 + v2t1S3(526) = −v−2z4t−1 − v−2z2t−1 + v−2t−1 + v−4z2t−1 − v−1z3t21 − v−1zt21S3(527) = v3z3t2−1 + v3zt2−1 − v2z4t−1t21 − v2z2t−1t21 + v4z2t1 + v2z4t1 + v2z2t1 + v2t1S3(527) = −v−2z4t−1 − v−2z2t−1 + v−2t−1 + v−4z2t−1 − v−1z3t21 − v−1zt21S3(528) = v−4z4t−1 + 3v−4z2t−1 + 2v−4t−1 − v−6t−1 − v−5z3t21 − v−5zt21S3(528) = v−4z4t−1t21 − v−4z2t−1t21 + vz3t41 + v−2z6t1 + 3v−2z4t1 + 2v−2z2t1 + 2v−2t1 − 2v−4z4t1 − v−4t1


S3(530) = −v3zt2−1 − v2z2t1 − z2t1 + v−2t1S3(530) = −v−2z2t2−1t1 + v2t−1 − z2t−1 + v−2z2t−1 + v−1zt21S3(531) = −vz3t2−1 + z4t−1t21 + z2t−1t21 − v2z2t1 − z4t1 + t1S3(531) = z2t2−1t1 + z4t−1 + t−1 − v−2z2t−1 + vz3t21S3(532) = −v−1zt−1t1 + v−3z3t−1t1 + v−3zt−1t1 − vz − vz−1 + v−1z3 + 3v−1z + v−1z−1 − v−3z3 − 2v−3zS3(532) = z2t3−1 + v−3z3t−1t1 − v−3zt−1t1 − v−5zt−1t1 + v−1z3 − v−3z3 − v−3z − v−3z−1 + v−5z + v−5z−1

S3(533) = −vzt2−1 + v−1z3t2−1 − v−3zt2−1 − v−2z2t1 + v−4t1S3(534) = v−3z3t2−1 + v−4z2t−1t21 + 2v−2z2t1 + 2v−2t1 − 2v−4z2t1 − v−4t1S3(534) = −v−1z3t4−1 − v−6z4t2−1t1 + 2v−6z2t2−1t1 − v−4z6t−1 − 2v−4z4t−1 + 3v−4z2t−1 + 2v−4t−1 + 2v−6z4t−1 −2v−6z2t−1 − v−6t−1 − v−5zt21S3(535) = v5z3t−1t1 + v5zt−1t1 + v3z3t−1t1 + v3zt−1t1 − v7z − v5z−1 + v3z + v3z−1

S3(535) = v−2z2t3−1 − v−5z3t2−1t21 − v−3z3t−1t1 − v−3zt−1t1 + 3v−5z3t−1t1 − v−5zt−1t1 + v−2z2t31 + 2v−3z3 −v−3z−1 − 2v−5z3 + v−5z−1

S3(536) = vzt−1t1 − v−1z3t−1t1 − v−1zt−1t1 + v3z − vz3 − 3vz − vz−1 + v−1z3 + 2v−1z + v−1z−1

S3(536) = v−4z2t3−1−v−5z3t−1t1−v−5zt−1t1−v−7zt−1t1−v−3z3−2v−3z−v−3z−1+v−5z3+v−5z+v−5z−1+v−7zS3(538) = −v3z3t−1t1 + v3zt−1t1 − v6z2t31 − v5z3 − v5z + v3z3 − v3z−1 + vz + vz−1

S3(539) = −v−5zt−1t1 − v−1z − v−1z−1 + v−3z−1 + v−5zS3(539) = v−1zt−1t1−2v−3z3t−1t1−2v−3zt−1t1−v2z2t31 +z4t31 +z2t31 +vz−2v−1z3−4v−1z−v−1z−1+2v−3z3+3v−3z + v−3z−1

S3(540) = v5z3t2−1 + 2v5zt2−1 + v4z4t1 + 3v4z2t1 + v4t1S3(540) = v−4z4t2−1t1 + 2v−4z2t2−1t1 − v−4z4t−1 − v−4z2t−1 + v−4t−1 − v−3z3t21 − 2v−3zt21S3(541) = v−8z4t2−1t1 + 2v−8z2t2−1t1 + v−6z6t−1 + 6v−6z4t−1 + 9v−6z2t−1 + 2v−6t−1 − 2v−8z4t−1 − 5v−8z2t−1 −v−8t−1 − v−7z3t21 − 2v−7zt21S3(542) = v5zt2−1 + v3zt2−1 + v4z2t1 + v4t1 + v2z2t1S3(542) = v−2z2t2−1t1 + v−4z2t2−1t1 − v−2z2t−1 − v−4z2t−1 + v−4t−1 − v−1zt21 − v−3zt21S3(543) = −v−3zt2−1+v−4z2t−1t21 −2v−6z2t−1t21 +v−1z3t41 +v−2z4t1+4v−2z2t1+2v−2t1−2v−4z4t1−7v−4z2t1−2v−4t1 + 4v−6z2t1 + v−6t1S3(543) = v−2z2t−1 + v−2t−1 + v−4z2t−1 + v−4t−1 − v−6t−1S3(544) = vz3t−1t1 + 2vzt−1t1 − v−1z3t−1t1 − v−1zt−1t1 − v3z−1 − vz3 − vz + vz−1 + v−1z3 + v−1zS3(544) = −v−2z4t3−1+v−4z2t3−1−v−5z3t2−1t21 −v−3z5t−1t1−4v−3z3t−1t1−2v−3zt−1t1+5v−5z3t−1t1+v−5zt−1t1−2v−7zt−1t1 + v−2z2t31 + 3v−3z3 + v−3z − 3v−5z3 − 3v−5z − v−5z−1 + 2v−7z + v−7z−1

S3(545) = v−1z3t3−1t1 − vzt2−1 − v−1z3t2−1 + v−1zt2−1 − v−2z2t−1t21 − z2t1 + t1 + v−2z2t1S3(545) = −z2t2−1t1 − v2z2t−1 + z2t−1 + v−2t−1 − vz3t21 − vzt21S3(546) = −v2z4t−1t21 − v2z2t−1t21 − v4t1 + v2z4t1 + 2v2z2t1 + 2v2t1S3(546) = −v−2z4t2−1t1+v−2z2t2−1t1−v−1z3t−1t31 −z4t−1+ t−1+v−2z4t−1−2v−2z2t−1−vz3t21 −vzt21 +3v−1z3t21 +v−1zt21S3(547) = v−8z8t3−1 + 5v−8z6t3−1 + 7v−8z4t3−1 + 3v−8z2t3−1 − v−13z3t2−1t21 − 3v−11z7t−1t1 − 15v−11z5t−1t1 −20v−11z3t−1t1 − 6v−11zt−1t1 + 4v−13z3t−1t1 + v−13zt−1t1 + v−10z2t31 − 3v−9z7 − 16v−9z5 − 24v−9z3 − 11v−9z −v−9z−1 + 3v−11z7 + 16v−11z5 + 26v−11z3 + 12v−11z + v−11z−1 − 2v−13z3 − v−13zS3(547) = −3v−9z5t−1t1 − 8v−9z3t−1t1 − 4v−9zt−1t1 + v−11zt−1t1 + v−6z6t31 + 3v−6z4t31 + 2v−6z2t31 − 3v−7z5 −10v−7z3 − 8v−7z − 2v−7z−1 + 3v−9z5 + 10v−9z3 + 9v−9z + 3v−9z−1 − v−11z − v−11z−1

S3(548) = −v3z3t2−1 + vzt2−1 + v4t1 − v2z4t1 − 2v2z2t1 − v2t1 + z2t1 + t1S3(548) = v3z3t4−1 −v−2z4t3−1t21 + z2t2−1t1 + 5v−2z4t2−1t1 − 2v−2z2t2−1t1 + 2v−1z3t−1t31 + z6t−1 + 5z4t−1 − z2t−1 −5v−2z4t−1 + 2v−2z2t−1 + v−2t−1 + 2vz3t21 − 5v−1z3t21 − v−1zt21S3(549) = v−6z2t2−1t1 + v−8z2t2−1t1 + v−4z4t−1 + 3v−4z2t−1 + v−6z4t−1 + 2v−6z2t−1 + 2v−6t−1 − 2v−8z2t−1 −


v−8t−1 − v−5zt21 − v−7zt21S3(549) = v−3zt2−1 − v−4z2t−1t21 + z2t1 + 2t1 − v−2z4t1 − 3v−2z2t1 − v−2t1 + 2v−4z2t1S3(550) = v4z2t−1 + v4t−1 + v3z3t21 + 2v3zt21S3(550) = −v−3z3t2−1 − 2v−3zt2−1 − v−5z3t2−1 − v−5zt2−1 + v−6z2t1 + v−6t1S3(552) = −v−2z4t2−1t1 − 2v−2z2t2−1t1 + z2t−1 + 2t−1 + v−2z4t−1 + v−2z2t−1 − v−2t−1 + v−1z3t21 + v−1zt21S3(552) = −v3z3t2−1 − v3zt2−1 − v2z2t−1t21 − v2z4t1 − v2z2t1 − v2t1 + z2t1 + 2t1S3(553) = −v−1z3t2−1 − v−1zt2−1 − v−2z2t−1t21 − z2t1 + 2v−2z2t1 + v−2t1S3(553) = v−8z4t2−1t1+2v−8z2t2−1t1+v−6z6t−1+6v−6z4t−1+ 10v−6z2t−1+4v−6t−1−2v−8z4t−1−6v−8z2t−1−3v−8t−1S3(554) = −v−2z4t−1 − 2v−2z2t−1 + v−4z2t−1 + v−4t−1 − v−1z3t21 − 2v−1zt21S3(554) = v3z3t2−1 + 2v3zt2−1 − v2z4t−1t21 − 2v2z2t−1t21 + v4z2t1 + v4t1 + v2z4t1 + 2v2z2t1S3(555) = −v2z2t2−1t1 − v4z2t−1 − v4t−1 + 2v2z2t−1 + v2t−1 + z2t−1 + t−1 − v3z3t21 − vz3t21 − vzt21S3(556) = v−4z4t−1 + 3v−4z2t−1 + v−4t−1 − v−5z3t21 − 2v−5zt21S3(556) = v−3zt2−1+v−4z4t−1t21 −v−4z2t−1t21 +vz3t41 +v−2z6t1+3v−2z4t1+2v−2z2t1+3v−2t1−2v−4z4t1−2v−4t1S3(557) = v4t−1 + v3zt21 + vzt21S3(557) = −v−1zt2−1 − v−3zt2−1 − v−5zt2−1 + v−6t1S3(558) = v3zt2−1 + vzt2−1 − v2z2t−1t21 − z2t−1t21 + v4t1 + v2z2t1 + z2t1S3(558) = −z2t−1 − v−2z2t−1 + v−4t−1 − vzt21 − v−1zt21S3(559) = −v−1zt2−1 − v−4z2t−1t21 + z2t1 + t1 − v−2z4t1 − 3v−2z2t1 − v−2t1 + 2v−4z2t1 + v−4t1S3(559) = v−6z2t2−1t1 + v−8z2t2−1t1 + v−4z4t−1 + 3v−4z2t−1 + v−4t−1 + v−6z4t−1 + 2v−6z2t−1 + 2v−6t−1 −2v−8z2t−1 − 2v−8t−1S3(560) = −v3zt2−1 − z2t−1t21 − v2z2t1 + v−2t1S3(560) = −z2t2−1t1 − v−2z2t2−1t1 + v2t−1 + v−2z2t−1 + v−1zt21S3(561) = v4z2t3−1 − vz3t2−1t21 + 2v3zt−1t1 + 2vz3t−1t1 − v5z−1 + v3z3 + v3z−1 − vz3

S3(561) = 2v−1z3t−1t1 − 2v−3zt−1t1 + v2z2t31 + vz3 − v−1z3 − v−3z−1 + v−5z−1

S3(562) = v−3z3t2−1t21 + v−1z5t−1t1 + 4v−1z3t−1t1 + 2v−1zt−1t1 − 4v−3z3t−1t1 − 2v−3zt−1t1 + z4t31 − vz − vz−1 −2v−1z3 + v−1z−1 + 2v−3z3 + v−3zS3(562) = −v−1z3t−1t1−v−1zt−1t1+2v−3z3t−1t1−v−5zt−1t1+z2t31 +2v−1z3+v−1z−2v−3z3−2v−3z−v−3z−1+v−5z + v−5z−1

S3(563) = −z2t2−1t1 − v2z2t−1 + v2t−1 + z2t−1 − vz3t21 + v−1zt21S3(563) = v−1z3t3−1t1 − vzt2−1 − v−1z3t2−1 − v−3zt2−1 − v−2z2t−1t21 − z2t1 + v−2z2t1 + v−4t1S3(564) = vz3t3−1t1 +v3z3t2−1 − 3vz3t2−1 −vzt2−1 + z4t−1t21 − z2t−1t21 +v2z4t1 −v2z2t1 −v2t1 − z4t1 + 2z2t1 + 2t1S3(564) = z4t2−1t1 + z2t2−1t1 − v−2z2t2−1t1 + v2t−1 − z4t−1 − 2z2t−1 + v−2z2t−1 + v−1zt21S3(565) = v−8z2t3−1 − 4v−9z5t−1t1 − 9v−9z3t−1t1 − 4v−9zt−1t1 + v−6z6t31 + 2v−6z4t31 + v−6z2t31 − 4v−7z5 −12v−7z3 − 8v−7z − v−7z−1 + 4v−9z5 + 12v−9z3 + 8v−9z + v−9z−1

S3(565) = v−8z8t3−1 + 5v−8z6t3−1 + 8v−8z4t3−1 + 4v−8z2t3−1 − v−13z3t2−1t21 − 3v−11z7t−1t1 − 14v−11z5t−1t1 −19v−11z3t−1t1 − 6v−11zt−1t1 + 4v−13z3t−1t1 + 2v−13zt−1t1 − 3v−9z7 − 15v−9z5 − 22v−9z3 − 11v−9z − 2v−9z−1 +3v−11z7 + 15v−11z5 + 24v−11z3 + 13v−11z + 3v−11z−1 − 2v−13z3 − 2v−13z − v−13z−1

S3(566) = v−2z2t−1 + v−4z2t−1 + v−4t−1 − v−3zt21 − v−5zt21S3(566) = v−5zt2−1+v−4z2t−1t21 −2v−6z2t−1t21 +v−1z3t41 +v−2z4t1+4v−2z2t1+3v−2t1−2v−4z4t1−7v−4z2t1−2v−4t1 + 4v−6z2t1S3(567) = −v−3z3t4−1 −v−5z3t4−1 +2v−8z2t2−1t1 + 3v−10z2t2−1t1 +2v−6z4t−1 + 5v−6z2t−1 +v−6t−1 + 3v−8z4t−1 +5v−8z2t−1 + 2v−8t−1 − 6v−10z2t−1 − 2v−10t−1 − v−9zt21S3(568) = −v−2z4t4−1t1+z2t3−1+3v−2z4t3−1+3v−5z3t2−1t21+3v−3z5t−1t1+9v−3z3t−1t1+v−3zt−1t1−10v−5z3t−1t1−2v−5zt−1t1 − v−2z2t31 − 2v−1z − v−1z−1 − 4v−3z3 + v−3z + v−3z−1 + 4v−5z3 + v−5z


S3(569) = −3v−7z3t−1t1−3v−7zt−1t1+v−4z4t31 +v−4z2t31 −3v−5z3−5v−5z−v−5z−1+3v−7z3+5v−7z+v−7z−1

S3(569) = v−12z4t4−1t1 + v−10z10t3−1 + 8v−10z8t3−1 + 22v−10z6t3−1 + 24v−10z4t3−1 + 7v−10z2t3−1 − 4v−12z4t3−1 −2v−12z2t3−1 − 2v−15z3t2−1t21 − 2v−13z9t−1t1 − 14v−13z7t−1t1 − 34v−13z5t−1t1 − 34v−13z3t−1t1 − 9v−13zt−1t1 +8v−15z3t−1t1+4v−15zt−1t1−2v−11z9−15v−11z7−38v−11z5−40v−11z3−16v−11z−2v−11z−1+2v−13z9+15v−13z7+38v−13z5 + 44v−13z3 + 20v−13z + 3v−13z−1 − 4v−15z3 − 4v−15z − v−15z−1

S3(570) = −v3z3t4−1 − v−2z4t2−1t1 + 2v−2z2t2−1t1 − z6t−1 − 3z4t−1 + t−1 + 2v−2z4t−1 − 2v−2z2t−1 + v−1z3t21S3(570) = v5zt2−1 − v3z3t2−1 − v3zt2−1 − v4t1 + 2v2z2t1 + 2v2t1S3(571) = −v4z2t3−1 + vz3t2−1t21 − 4vz3t−1t1 − v4z2t31 − 2v3z3 − v3z + 2vz3 − vz−1 + v−1z + v−1z−1

S3(572) = v5z3t2−1 + v5zt2−1 − v6t1 + v4z4t1 + 3v4z2t1 + 2v4t1S3(572) = −vz3t4−1 +v−4z4t3−1t21 − 5v−4z4t2−1t1 + 3v−4z2t2−1t1 − 2v−3z3t−1t31 −v−2z6t−1 − 5v−2z4t−1 +v−2t−1 +5v−4z4t−1 − 3v−4z2t−1 − 2v−1z3t21 − v−1zt21 + 5v−3z3t21S3(573) = v−8z8t3−1+6v−8z6t3−1+10v−8z4t3−1+4v−8z2t3−1−v−10z2t3−1−2v−11z7t−1t1−11v−11z5t−1t1−16v−11z3t−1t1−6v−11zt−1t1 + 2v−13zt−1t1 − 2v−9z7 − 12v−9z5 − 21v−9z3 − 12v−9z − 2v−9z−1 + 2v−11z7 + 12v−11z5 + 21v−11z3 +14v−11z + 3v−11z−1 − 2v−13z − v−13z−1

S3(573) = −2v−9z5t−1t1 − 7v−9z3t−1t1 − 4v−9zt−1t1 + v−6z6t31 + 4v−6z4t31 + 3v−6z2t31 − 2v−7z5 − 8v−7z3 −7v−7z − v−7z−1 + 2v−9z5 + 8v−9z3 + 7v−9z + v−9z−1

S3(574) = −v−5z5t4−1 − 2v−5z3t4−1 + 3v−10z4t2−1t1 + 5v−10z2t2−1t1 + 3v−8z6t−1 + 14v−8z4t−1 + 17v−8z2t−1 +4v−8t−1 − 6v−10z4t−1 − 12v−10z2t−1 − 3v−10t−1 − v−9z3t21 − v−9zt21S3(575) = −2z2t2−1t1 + vz3t−1t31 − 2v2z2t−1 − v2t−1 + 4z2t−1 + 2t−1 − 3vz3t21 − vzt21S3(576) = z2t3−1 − v−2z4t3−1 − v−2z2t3−1 − 2v−3zt−1t1 + 2v−5z3t−1t1 + v−5zt−1t1 − 2v−1z − v−1z−1 + 2v−3z3 +4v−3z + v−3z−1 − 2v−5z3 − 2v−5zS3(577) = −v−1z3t4−1 − v−6z4t2−1t1 + 2v−6z2t2−1t1 − v−4z6t−1 − 2v−4z4t−1 + 4v−4z2t−1 + 3v−4t−1 + 2v−6z4t−1 −3v−6z2t−1 − 2v−6t−1 + v−5z3t21S3(578) = z2t2−1t1 − 2z2t−1 + v−2t−1 − vzt21S3(579) = −v6t−1 + v4z2t−1 + v4t−1 + v2z2t−1 + v2t−1S3(579) = −v−5z3t3−1t1−v−3z3t2−1−v−3zt2−1+2v−5z3t2−1+2v−6z2t−1t21 +3v−4z2t1+2v−4t1−3v−6z2t1−v−6t1S3(580) = v−4z4t−1 + 4v−4z2t−1 + 3v−4t−1 − v−6z2t−1 − 2v−6t−1S3(580) = −v−3z3t2−1 − v−3zt2−1 + v−4z4t−1t21 − v−4z2t−1t21 + vz3t41 + v−2z6t1 + 3v−2z4t1 + v−2z2t1 + v−2t1 −2v−4z4t1 + v−4z2t1S3(581) = −v−5z5t4−1 − 2v−5z3t4−1 + 3v−10z4t2−1t1 + 5v−10z2t2−1t1 + 3v−8z6t−1 + 14v−8z4t−1 + 17v−8z2t−1 +3v−8t−1 − 6v−10z4t−1 − 12v−10z2t−1 − 2v−10t−1 − v−9z3t21 − 2v−9zt21S3(582) = −v−3z3t4−1−v−5z3t4−1+2v−8z2t2−1t1+3v−10z2t2−1t1+2v−6z4t−1+5v−6z2t−1+3v−8z4t−1+5v−8z2t−1+3v−8t−1 − 6v−10z2t−1 − 2v−10t−1 − v−7zt21 − v−9zt21S3(583) = −vz3t2−1 + z4t−1t21 − 2z2t−1t21 + v5z3t41 + v2z6t1 + 3v2z4t1 − 2z4t1 + 2z2t1 + t1S3(583) = v2t−1 − 2z2t−1 − t−1 + v−2t−1 − v3zt21 + vz3t21 + vzt21S3(584) = −3v−7z3t−1t1 − 2v−7zt−1t1 + v−4z4t31 + v−4z2t31 − 3v−5z3 − 5v−5z − 2v−5z−1 + 3v−7z3 + 5v−7z +3v−7z−1 − v−9z−1

S3(584) = v−12z4t4−1t1 + v−10z10t3−1 + 8v−10z8t3−1 + 22v−10z6t3−1 + 24v−10z4t3−1 + 8v−10z2t3−1 − 4v−12z4t3−1 −2v−12z2t3−1 − 2v−15z3t2−1t21 − 2v−13z9t−1t1 − 14v−13z7t−1t1 − 34v−13z5t−1t1 − 33v−13z3t−1t1 − 8v−13zt−1t1 +8v−15z3t−1t1+2v−15zt−1t1+v−12z2t31 −2v−11z9− 15v−11z7−38v−11z5−39v−11z3− 14v−11z−v−11z−1+2v−13z9+15v−13z7 + 38v−13z5 + 43v−13z3 + 16v−13z + v−13z−1 − 4v−15z3 − 2v−15zS3(585) = −v−5z5t6−1 + 5v−14z4t4−1t1 + 5v−12z10t3−1 + 30v−12z8t3−1 + 64v−12z6t3−1 + 55v−12z4t3−1 + 10v−12z2t3−1 −20v−14z4t3−1 − 4v−14z2t3−1 − 6v−17z3t2−1t21 − 10v−15z9t−1t1 − 46v−15z7t−1t1 − 76v−15z5t−1t1 − 64v−15z3t−1t1 −12v−15zt−1t1 + 24v−17z3t−1t1 +6v−17zt−1t1 +v−14z2t31 − 10v−13z9 − 51v−13z7 −86v−13z5 − 70v−13z3 − 21v−13z−2v−13z−1 + 10v−15z9 + 51v−15z7 + 86v−15z5 + 82v−15z3 + 27v−15z + 3v−15z−1 − 12v−17z3 − 6v−17z − v−17z−1


S3(587) = −v−3z3t2−1 − v−3zt2−1 − v−5z3t2−1 − 2v−5zt2−1 + v−4z2t1 + v−6z2t1 + v−6t1S3(588) = −v−3z3t2−1 − v−3zt2−1 − v−5z3t2−1 − v−5zt2−1 + v−4z2t1 + v−4t1 + v−6z2t1S3(592) = v5zt2−1 − v6t1 + 2v4z2t1 + 2v4t1 + v2z2t1S3(592) = v−4z2t2−1t1 + v−2z2t−1 + 2v−4t−1 − v−6t−1 − v−3zt21S3(593) = −v−5z5t2−1 − 4v−5z3t2−1 − 3v−5zt2−1 + v−6z4t1 + 3v−6z2t1 + v−6t1

Knots in L(4, 1)

S3(22) = v−2t−1 − v−1zt1t2S3(23) = t−1S3(23) = v−2z2t3−1 − 2v−5zt−1t2 + 2v−4z2t1 + v−4t1S3(32) = v−6z2t2−1t2 − 2v−5z3t−1t1 − v−5zt−1t1 − v−3zt22 − v−3z3 − v−3z − v−3z−1 + v−5z3 + v−5z + v−5z−1

S3(32) = −v−3zt−1t1 − v−1z − v−1z−1 + v−3z + v−3z−1

S3(35) = −v−1zt22 − v−1z−1 + v−3z−1

S3(35) = v−6z2t2−1t2 − 2v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 2v−3z − v−3z−1 + v−5z3 + 2v−5z + v−5z−1

S3(36) = −v−3z3t4−1 + 3v−8z2t2−1t2 −6v−7z3t−1t1 − 2v−7zt−1t1 − v−5zt22 − 3v−5z3 − 2v−5z− v−5z−1 + 3v−7z3 +2v−7z + v−7z−1

S3(41) = −v−3z3t3−1t1 + 2v−3z3t2−1 − v−3zt2−1 + 2v−4z2t−1t1t2 − 2v−3z3t21 − v−3zt21 − v−4z2t2 + v−4t2S3(42) = −v−9z3t3−1t2+3v−8z4t2−1t1+v−8z2t2−1t1+2v−8z2t−1t22+v−6z6t−1+6v−6z4t−1+5v−6z2t−1+2v−6t−1−3v−8z4t−1 − 2v−8z2t−1 − v−8t−1 − 4v−7z3t1t2 − 2v−7zt1t2S3(42) = v−2z2t1 + 2v−2t1 − v−4t1S3(44) = vzt2−1 − z2t−1t1t2 − vzt21 + z2t2 + t2S3(44) = v−1z3t2−1 + v−2z2t−1t1t2 − v−1z3t21 − v−1zt21 − v−2z2t2 + v−2t2S3(45) = −2v−7zt−1t2 + v−2z2t1t22 + 2v−4z2t1 + v−4t1 + 2v−6z2t1S3(45) = v−6z2t−1t21 + v−4z4t1 + 4v−4z2t1 + 3v−4t1 − 2v−6z2t1 − 2v−6t1S3(48) = −v−5z3t−1t2 − 2v−5zt−1t2 + v−4z4t1 + 3v−4z2t1 + v−4t1S3(48) = z2t−1 + t−1 − v2z2t31 − vz3t1t2S3(411) = t−1 − z2t31 − vzt1t2 + v−1zt1t2S3(411) = −v−3zt−1t2 − v−5zt−1t2 + v−2z2t1 + v−4z2t1 + v−4t1S3(412) = vzt2−1 − z2t−1t1t2 + v2t2 + z2t2S3(412) = v−1z3t2−1 − v−3zt2−1 + v−2z2t−1t1t2 − v−1z3t21 − v−1zt21 − v−2z2t2 + v−4t2S3(414) = −v−7zt−1t2 + v−2z2t1t22 + 2v−4z2t1 + 2v−4t1 + v−6z2t1 − v−6t1S3(415) = −vz3t−1t1 + v−1zt−1t1 − v3zt22 + 2v3z − 2vz − vz−1 + v−1z−1

S3(415) = v−1z3t2−1t21 −v−2z2t2−1t2−vzt−1t1−v−1z3t−1t1−v2z2t21 t2+vzt22−vz3−2vz−vz−1+v−1z3+2v−1z+v−1z−1

S3(416) = v2t−1 + vzt1t2S3(416) = v−2z2t3−1 + v−4z2t3−1 − v−5zt−1t2 − 2v−7zt−1t2 + v−4z2t1 + 2v−6z2t1 + v−6t1S3(417) = v−1zt2−1 + v−1z3t−1t31 − 2v−2z2t−1t1t2 − vzt21 + t2 + v−2z2t2S3(417) = −vzt2−1 − z2t2 + v−2t2S3(418) = −v−5zt−1t2 + v−2z2t1 + v−2t1 + v−4z2t1S3(418) = −v−9z3t3−1t2+3v−8z4t2−1t1+2v−8z2t2−1t1+v−8z2t−1t22+v−6z6t−1+6v−6z4t−1+7v−6z2t−1+3v−6t−1−3v−8z4t−1 − 4v−8z2t−1 − 2v−8t−1 − 2v−7z3t1t2 − v−7zt1t2S3(419) = v−2z2t2−1t2 − v−1z3t−1t1 − vzt22 + 2vz − 2v−1z − v−1z−1 + v−3z−1


S3(419) = −vz3t−1t1 − v4z2t21 t2 + v3zt22 − v3z3 − 2v3z − v3z−1 + vz3 + 2vz + vz−1

S3(420) = v3z3t−1t1 + v3zt−1t1 + v5zt22 − 2v5z − v5z−1 + 2v3z + v3z−1

S3(420) = −v−3z3t2−1t21 + v−4z2t2−1t2 + v−3z3t−1t1 − v−3zt−1t1 + z2t21 t2 − v−1zt22 + v−1z3 + 2v−1z − v−3z3 −2v−3z − v−3z−1 + v−5z−1

S3(421) = v−6z2t−1t21 − v−7zt−1t2 + v−4z4t1 + 4v−4z2t1 + 2v−4t1 − v−6z2t1 − v−6t1S3(422) = v−4z4t3−1 + 2v−4z2t3−1 − 2v−7z3t−1t2 − 3v−7zt−1t2 + 2v−6z4t1 + 4v−6z2t1 + v−6t1S3(424) = v−4z4t5−1 − 4v−11z3t3−1t2 + 12v−10z4t2−1t1 + 3v−10z2t2−1t1 + 3v−10z2t−1t22 + 4v−8z6t−1 + 18v−8z4t−1 +10v−8z2t−1 + 3v−8t−1 − 12v−10z4t−1 − 6v−10z2t−1 − 2v−10t−1 − 6v−9z3t1t2 − 2v−9zt1t2S3(427) = −v−2z2t−1 + v−2t−1 + z2t31 + v−1z3t1t2 − v−1zt1t2S3(52) = v3zt2−1 − v2z2t−1t1t2 + 2v2z2t2 + v2t2S3(52) = −v−3zt2−1 + v−4z2t−1t1t2 − v−3z3t21 − v−3zt21 + v−4t2S3(54) = v−8z4t2−1t2 + 2v−8z2t2−1t2 − 2v−7z5t−1t1 − 5v−7z3t−1t1 − 2v−7zt−1t1 − v−5z3t22 − v−5zt22 − v−5z5 −3v−5z3 − 3v−5z − v−5z−1 + v−7z5 + 3v−7z3 + 3v−7z + v−7z−1

S3(54) = −v−1z3t−1t1 − v2z2t21 t2 − vz3 − vz − vz−1 + v−1z3 + v−1z + v−1z−1

S3(55) = −v4t−1 + v2z2t−1 + 2v2t−1S3(55) = v−4z4t4−1t1 − 3v−4z4t3−1 + v−4z2t3−1 − 3v−7z3t2−1t1t2 + 6v−6z4t−1t21 + 2v−6z2t−1t21 + 4v−7z3t−1t2 −2v−7zt−1t2 + v−2z2t1t22 + 2v−4z4t1 + 2v−4z2t1 + v−4t1 − 7v−6z4t1 − v−6z2t1S3(56) = v−12z4t4−1t2 − 4v−11z5t3−1t1 − v−11z3t3−1t1 − 3v−13z3t2−1t22 − v−9z9t2−1 − 8v−9z7t2−1 − 21v−9z5t2−1 −13v−9z3t2−1 − 3v−9zt2−1 + 6v−11z5t2−1 + 3v−11z3t2−1 + v−11zt2−1 + 12v−12z4t−1t1t2 + 4v−12z2t−1t1t2 + v−9z5t21 −6v−11z5t21 −4v−11z3t21 −v−11zt21 +v−6z2t32+v−10z8t2+7v−10z6t2+15v−10z4t2+11v−10z2t2+3v−10t2−6v−12z4t2−4v−12z2t2 − 2v−12t2S3(57) = −v−3z3t2−1t21 + v−4z2t2−1t2 − 2v−3zt−1t1 − v−3z−1 + v−5z−1

S3(58) = −z2t2−1t1 − v2z2t−1 − v2t−1 + 2z2t−1 + 2t−1 − vz3t1t2 − vzt1t2S3(58) = −v−2z4t3−1 −v−5z3t2−1t1t2 +2v−4z4t−1t21 +v−4z2t−1t21 + 3v−5z3t−1t2 −v−5zt−1t2 + z2t1t22 +v−2z2t1 +v−2t1 − 4v−4z4t1 − v−4z2t1S3(59) = −v−11z3t2−1t22 − v−7z7t2−1 − 5v−7z5t2−1 − 5v−7z3t2−1 − v−7zt2−1 + 4v−10z4t−1t1t2 + 2v−10z2t−1t1t2 +v−7z5t21 − 2v−9z5t21 − 2v−9z3t21 − v−9zt21 + v−4z2t32 + 2v−8z6t2 + 6v−8z4t2 + 8v−8z2t2 + 3v−8t2 − 2v−10z4t2 −2v−10z2t2 − 2v−10t2S3(59) = −v−5z5t21 − 4v−5z3t21 − 4v−5zt21 + v−7zt21 + v−6z4t2 + 2v−6z2t2 + v−6t2S3(512) = −z2t−1t22 − v2t−1 + z2t−1 + 2t−1 − vzt1t2S3(512) = −v−2z4t3−1 − v−5z3t2−1t1t2 + 2v−4z4t−1t21 + 2v−4z2t−1t21 + 2v−5z3t−1t2 − v−5zt−1t2 + v−2z4t1 +2v−2z2t1 + v−2t1 − 3v−4z4t1 − 2v−4z2t1S3(513) = v−1z3t2−1t21 − v−2z2t2−1t2 − vzt−1t1 + v−1zt−1t1 − vz−1 + v−1z−1

S3(514) = −v−2z2t2−1t1 + v−1z3t−1t21 t2 − v−2z2t−1t22 − z2t−1 + 2v−2z2t−1 + v−2t−1 − vzt1t2 − v−1z3t1t2S3(514) = v−2z2t−1t21 + v−3z3t−1t2 − v−5zt−1t2 + z2t1 + t1 − v−2z4t1 − 2v−2z2t1 − v−2t1 + v−4z2t1 + v−4t1S3(515) = −v−11z3t2−1t22 − v−7z7t2−1 − 5v−7z5t2−1 − 6v−7z3t2−1 − 2v−7zt2−1 + 4v−10z4t−1t1t2 + 3v−10z2t−1t1t2 −2v−9z5t21 − 3v−9z3t21 − v−9zt21 + 2v−8z6t2 + 8v−8z4t2 + 8v−8z2t2 + 2v−8t2 − 2v−10z4t2 − 3v−10z2t2 − v−10t2S3(515) = v−8z2t−1t1t2−v−5z5t21−3v−5z3t21−2v−5zt21−v−7z3t21+2v−6z4t2+4v−6z2t2+2v−6t2−v−8z2t2−v−8t2S3(516) = z4t2−1t1 + z2t−1t22 − 2v2z2t−1 − z4t−1 + v−2t−1 − 2vz3t1t2 − vzt1t2S3(516) = −v2z2t3−1 + v−1z3t2−1t1t2 − z4t−1t21 − 2v−1z3t−1t2 + v−1zt−1t2 − v4z2t1t22 + v2z4t1 + 2v2z2t1 + v2t1 +2z4t1 − z2t1S3(517) = v−6z2t2−1t2 + v−8z2t2−1t2 − 2v−5z3t−1t1 − v−5zt−1t1 − 2v−7z3t−1t1 − v−7zt−1t1 − v−5zt22 − v−3z3 −v−3z − v−5z−1 + v−7z3 + v−7z + v−7z−1

S3(517) = −v−1zt−1t1 + v−3zt−1t1 − z2t21 t2 − vz − vz−1 + 2v−1z + v−1z−1 − v−3zS3(519) = −v−2z2t2−1t1 + v−1z3t−1t21 t2 − v−2z2t−1t22 − z2t−1 − t−1 + 2v−2z2t−1 + 2v−2t−1 + z2t31 − v−1z3t1t2 −


2v−1zt1t2S3(519) = v−2z2t−1t21 + v−3z3t−1t2 + z2t1 + t1 − v−2z4t1 − 2v−2z2t1S3(520) = −v−7zt2−1+v−8z2t−1t1t2−v−5z5t21 −3v−5z3t21 −v−5zt21 −v−7z3t21 +2v−6z4t2+5v−6z2t2+2v−6t2−v−8z2t2 − v−8t2S3(520) = −v−9z3t3−1t1−v−7z7t2−1−6v−7z5t2−1−11v−7z3t2−1−5v−7zt2−1+3v−9z3t2−1+2v−9zt2−1+v−10z2t−1t1t2−v−9z3t21 + v−8z6t2 + 4v−8z4t2 + 5v−8z2t2 + 2v−8t2 − v−10z2t2 − v−10t2S3(521) = −v−2z4t2−1t1 − v−1z3t−1t21 t2 + v−2z2t−1t22 − z4t−1 − 2z2t−1 + v−2z4t−1 − v−2z2t−1 + v−4t−1 + z4t31 +z2t31 − 2v−1zt1t2S3(521) = −v2z4t−1t21 − v2z2t−1t21 + 2vzt−1t2 − v6z2t1t22 + v4z4t1 + 2v4z2t1 + v4t1 + v2z4t1 − v2z2t1S3(522) = −vzt2−1 + v−1zt21 − z2t2 + t2S3(522) = v−1z3t3−1t1 − vzt2−1 − 2v−1z3t2−1 − v−1zt2−1 − 2v−2z2t−1t1t2 + 2v−1z3t21 + v−1zt21 + v−2z2t2 + v−2t2S3(524) = v−1zt−1t1 − vzt22 + vz − vz−1 − v−1z + v−1z−1

S3(524) = −v−3z3t4−1 + 3v−8z2t2−1t2 −6v−7z3t−1t1 − 3v−7zt−1t1 − 3v−5z3 − 3v−5z− v−5z−1 + 3v−7z3 + 3v−7z+v−7z−1

S3(525) = v5zt2−1 + v3z3t21 + v3zt21 + 2v4z2t2 + v4t2S3(525) = −v−5z3t3−1t1−v−3z3t2−1−v−3zt2−1+v−5z3t2−1−v−5zt2−1+2v−6z2t−1t1t2−2v−5z3t21 −v−5zt21 +v−6t2S3(528) = −v−5zt−1t1 − v−3z3t22 − v−3zt22 − 2v−3z − v−3z−1 + 2v−5z + v−5z−1

S3(528) = −2v−3z3t−1t1−v−3zt−1t1+vz3t41 +z4t21 t2−z2t21 t2−2v−1z3−2v−1z−v−1z−1+2v−3z3+2v−3z+v−3z−1

S3(532) = t−1 − v−2z2t−1 − vzt1t2 + v−1z3t1t2 + v−1zt1t2S3(532) = z2t3−1 + v−3z3t−1t2 − v−3zt−1t2 − v−5zt−1t2 − v−2z4t1 + v−4z2t1 + v−4t1S3(534) = −v−3zt−1t1 + z2t21 t2 + v−1z3t22 − v−1z−1 + v−3z−1

S3(534) = −v−1z3t4−1 − v−6z4t2−1t2 + 2v−6z2t2−1t2 + 2v−5z5t−1t1 − 2v−5z3t−1t1 − v−5zt−1t1 − v−3zt22 + v−3z5 −v−3z−1 − v−5z5 + v−5z−1

S3(535) = v5z3t−1t1 + v3z3t−1t1 + v3zt−1t1 + v7zt22 − 2v7z + v5z − v5z−1 + v3z + v3z−1

S3(535) = −v−5z3t2−1t21 + v−6z2t2−1t2 − v−3z3t−1t1 − v−3zt−1t1 + v−5z3t−1t1 + v−2z2t21 t2 − v−3zt22 + v−3z3 +v−3z − v−3z−1 − v−5z3 − v−5z + v−5z−1

S3(536) = z2t−1 + t−1 + v3zt1t2 − vz3t1t2 − vzt1t2S3(536) = v−4z2t3−1 − v−5z3t−1t2 − v−5zt−1t2 − v−7zt−1t2 + v−4z4t1 + 2v−4z2t1 + v−4t1 + v−6z2t1S3(538) = −v3z3t−1t1 − v6z2t21 t2 + v5zt22 − v5z3 − 2v5z + v3z3 + v3z − v3z−1 + vz + vz−1

S3(538) = v−4z2t2−1t2 − v−3z3t−1t1 − v−1zt22 + v−1z − v−1z−1 − v−3z + v−3z−1

S3(539) = v−2z2t−1 + v−2t−1 − v2z2t31 + z4t31 + z2t31 + vzt1t2 − 2v−1z3t1t2 − 2v−1zt1t2S3(541) = v−8z4t2−1t2 + 2v−8z2t2−1t2 − 2v−7z5t−1t1 − 5v−7z3t−1t1 − v−7zt−1t1 − v−5z3t22 − 2v−5zt22 − v−5z5 −3v−5z3 − 2v−5z − v−5z−1 + v−7z5 + 3v−7z3 + 2v−7z + v−7z−1

S3(541) = −v−1z3t−1t1 − v−1zt−1t1 − v2z2t21 t2 + vzt22 − vz3 − 2vz − vz−1 + v−1z3 + 2v−1z + v−1z−1

S3(543) = −v−3zt−1t1+v−5zt−1t1+v−1z3t41 +z2t21 t2−2v−2z2t21 t2−v−1zt22−v−1z−v−1z−1+2v−3z+v−3z−1−v−5zS3(543) = −v−3zt−1t1 − v−5zt−1t1 − v−1z − v−3z−1 + v−5z + v−5z−1

S3(544) = −z2t2−1t1 − v2z2t−1 + 2z2t−1 + t−1 − vz3t1t2S3(544) = −v−2z4t3−1 + v−4z2t3−1 − v−5z3t2−1t1t2 + 2v−4z4t−1t21 + v−4z2t−1t21 + 3v−5z3t−1t2 − 2v−7zt−1t2 +z2t1t22 + v−2z2t1 + v−2t1 − 4v−4z4t1 − 2v−4z2t1 − v−4t1 + 2v−6z2t1 + v−6t1S3(546) = −v2z4t−1t21 + vzt−1t2 − v6z2t1t22 + v4z4t1 + 2v4z2t1 + v2z4t1 − v2z2t1 + v2t1S3(546) = −v−2z4t2−1t1−v−1z3t−1t21 t2+v−2z2t−1t22−z4t−1−2z2t−1+v−2z4t−1+v−2t−1+z4t31 +z2t31 −v−1zt1t2S3(547) = −v−11z3t2−1t22−v−7z7t2−1−5v−7z5t2−1−6v−7z3t2−1−3v−7zt2−1+v−9zt2−1+4v−10z4t−1t1t2+v−10z2t−1t1t2+v−7z5t21−2v−9z5t21−v−9z3t21+v−4z2t32+2v−8z6t2+5v−8z4t2+6v−8z2t2+3v−8t2−2v−10z4t2−v−10z2t2−2v−10t2S3(547) = −v−7zt2−1+v−8z2t−1t1t2−v−5z5t21 −3v−5z3t21 −2v−5zt21 −v−7z3t21 +2v−6z4t2+4v−6z2t2+v−6t2−v−8z2t2


S3(548) = −vz3t−1t2 − vzt−1t2 + v−1zt−1t2 − v2z2t1 + t1S3(548) = −v−2z4t3−1t21 +v−3z3t3−1t2+ z2t2−1t1+2v−2z4t2−1t1+2v−1z3t−1t21 t2−2v−2z2t−1t22 +2z4t−1+3z2t−1+t−1 − 2v−2z4t−1 − 2v−2z2t−1 − 2z4t31 − z2t31 − vzt1t2 + v−1z3t1t2 + v−1zt1t2S3(549) = v−6z2t2−1t2+v−8z2t2−1t2−2v−5z3t−1t1−2v−7z3t−1t1−v−7zt−1t1−v−3zt22 −v−5zt22 −v−3z3−v−5z−v−5z−1 + v−7z3 + v−7z + v−7z−1

S3(549) = −v−1zt−1t1 − z2t21 t2 + v−1zt22 − vz − vz−1 + v−1z + v−1z−1

S3(553) = v−1zt−1t1 − v2z2t21 t2 − vz3t22 − vzt22 − vz−1 + v−1z−1

S3(553) = v−8z4t2−1t2 + 2v−8z2t2−1t2 − 2v−7z5t−1t1 − 6v−7z3t−1t1 − 3v−7zt−1t1 − v−5z5 − 4v−5z3 − 4v−5z −v−5z−1 + v−7z5 + 4v−7z3 + 4v−7z + v−7z−1

S3(555) = v−3z3t2−1 + v−4z2t−1t1t2 − v−3z3t21 − v−3zt21 + v−2z2t2 + v−2t2 − v−4z2t2S3(555) = v3zt2−1 − v2z2t−1t1t2 − vz3t21 − vzt21 + v2z2t2 + z2t2 + t2S3(556) = −v−3z3t22 − 2v−3zt22 − v−3z − v−3z−1 + v−5z + v−5z−1

S3(556) = −2v−3z3t−1t1 − 2v−3zt−1t1 + vz3t41 + z4t21 t2 − z2t21 t2 + v−1zt22 − 2v−1z3 − 3v−1z − v−1z−1 + 2v−3z3 +3v−3z + v−3z−1

S3(559) = v−3zt−1t1 − z2t21 t2 − vzt22 − vz−1 + v−1z + v−1z−1 − v−3zS3(559) = v−6z2t2−1t2 + v−8z2t2−1t2 − 2v−5z3t−1t1 − v−5zt−1t1 − 2v−7z3t−1t1 − 2v−7zt−1t1 − v−3z3 − v−3z −v−5z − v−5z−1 + v−7z3 + 2v−7z + v−7z−1

S3(561) = v−2z2t2−1t2 − 2v−3zt−1t1 − v−3z−1 + v−5z−1

S3(562) = −2v−2z2t2−1t1 + v−1z3t−1t21 t2 − 2z2t−1 + 3v−2z2t−1 + v−2t−1 − vzt1t2 − 2v−1z3t1t2S3(562) = 2v−3z3t−1t2 − v−5zt−1t2 + v2z2t1t22 − z4t1 + t1 − 2v−2z4t1 − v−2z2t1 − v−2t1 + v−4z2t1 + v−4t1S3(564) = −v2z2t3−1+v−1z3t2−1t1t2−z4t−1t21 −vzt−1t2−2v−1z3t−1t2+v−1zt−1t2−v4z2t1t22+v2z4t1+2v2z2t1+2z4t1 − z2t1 + t1S3(564) = z4t2−1t1 − v−2z2t2−1t1 + z2t−1t22 − 2v2z2t−1 − z4t−1 + t−1 + v−2z2t−1 − 2vz3t1t2 − vzt1t2 + v−1zt1t2S3(565) = −2v−5z3t21 − v−5zt21 + v−2z2t32 + 4v−6z2t2 + 3v−6t2 − 2v−8t2S3(565) = −v−11z3t2−1t22−v−7z7t2−1−5v−7z5t2−1−7v−7z3t2−1−4v−7zt2−1+v−9zt2−1+4v−10z4t−1t1t2+2v−10z2t−1t1t2−2v−9z5t21 − 2v−9z3t21 − v−9zt21 + 2v−8z6t2 + 7v−8z4t2 + 6v−8z2t2 + v−8t2 − 2v−10z4t2 − 2v−10z2t2S3(566) = −v−1zt22 − v−3zt22 − v−3z−1 + v−5z−1

S3(566) = −2v−3zt−1t1 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 + v−3zt22 − 2v−1z − v−1z−1 + 2v−3z + v−3z−1

S3(567) = −v−3z3t4−1 − v−5z3t4−1 + 2v−8z2t2−1t2 + 3v−10z2t2−1t2 − 4v−7z3t−1t1 − v−7zt−1t1 − 6v−9z3t−1t1 −2v−9zt−1t1 − v−7zt22 − 2v−5z3 − v−5z − v−7z3 − v−7z − v−7z−1 + 3v−9z3 + 2v−9z + v−9z−1

S3(568) = −v−2z4t4−1t1 + z2t3−1 + 3v−2z4t3−1 + 3v−5z3t2−1t1t2 − 6v−4z4t−1t21 − 2v−4z2t−1t21 − 2v−3zt−1t2 −4v−5z3t−1t2 − z2t1t22 − 2v−2z4t1 + 7v−4z4t1 + 3v−4z2t1 + v−4t1S3(568) = v2t−1 − z2t−1 − t−1 + v−2t−1S3(569) = −v−3z3t21 − v−3zt21 + 2v−4z2t2 + 2v−4t2 − v−6t2S3(569) = v−12z4t4−1t2 − 4v−11z5t3−1t1 − 2v−11z3t3−1t1 − 2v−13z3t2−1t22 − v−9z9t2−1 − 8v−9z7t2−1 − 22v−9z5t2−1 −18v−9z3t2−1 − 5v−9zt2−1 + 6v−11z5t2−1 + 6v−11z3t2−1 + 2v−11zt2−1 + 8v−12z4t−1t1t2 + 4v−12z2t−1t1t2 − 4v−11z5t21 −4v−11z3t21 −v−11zt21 +v−10z8t2+6v−10z6t2+ 14v−10z4t2+ 10v−10z2t2+2v−10t2−4v−12z4t2−4v−12z2t2−v−12t2S3(570) = −v−3z3t3−1t2 + 2v−2z4t2−1t1 + 2v−2z2t−1t22 − 4z2t−1 − v−2z4t−1 + 2v−2z2t−1 + v−2t−1 − 3v−1z3t1t2 −v−1zt1t2S3(570) = −v4z2t3−1 − vz3t−1t2 + vzt−1t2 + v2z4t1 + v2z2t1 + v2t1S3(571) = vz3t2−1t21 −z2t2−1t2−2vz3t−1t1−vzt−1t1−v4z2t21 t2+v3zt22−v3z3−2v3z+vz3+vz−vz−1+v−1z+v−1z−1

S3(572) = v−4z4t3−1t21 − v−5z3t3−1t2 − 2v−4z4t2−1t1 + v−4z2t2−1t1 − 2v−3z3t−1t21 t2 + 2v−4z2t−1t22 − 2v−2z4t−1 −4v−2z2t−1 + 2v−4z4t−1 + v−4z2t−1 + v−4t−1 + 2v−2z4t31 + v−2z2t31 − v−3z3t1t2 − 2v−3zt1t2S3(573) = −v−9z3t3−1t1−v−7z7t2−1−6v−7z5t2−1−10v−7z3t2−1−3v−7zt2−1+3v−9z3t2−1+v−9zt2−1+2v−10z2t−1t1t2−2v−9z3t21 − v−9zt21 + v−8z6t2 + 5v−8z4t2 + 7v−8z2t2 + 2v−8t2 − 2v−10z2t2 − v−10t2


S3(573) = −v−5z5t21 − 4v−5z3t21 − 3v−5zt21 + v−7zt21 + v−6z4t2 + 3v−6z2t2 + 2v−6t2 − v−8t2S3(574) = −v−5z5t4−1 − 2v−5z3t4−1 + 3v−10z4t2−1t2 + 5v−10z2t2−1t2 − 6v−9z5t−1t1 − 12v−9z3t−1t1 − 3v−9zt−1t1 −v−7z3t22 − v−7zt22 − 3v−7z5 − 7v−7z3 − 4v−7z − v−7z−1 + 3v−9z5 + 7v−9z3 + 4v−9z + v−9z−1

S3(575) = vzt2−1 + vz3t−1t31 − 2z2t−1t1t2 − vz3t21 − vzt21 + 2z2t2 + t2S3(576) = z2t3−1−v−2z4t3−1−v−2z2t3−1−2v−3zt−1t2+2v−5z3t−1t2+v−5zt−1t2+2v−2z2t1+v−2t1−2v−4z4t1−2v−4z2t1S3(577) = −v−1z3t4−1−v−6z4t2−1t2+2v−6z2t2−1t2+2v−5z5t−1t1−3v−5z3t−1t1−2v−5zt−1t1+v−3z3t22 +v−3z5−v−3z3 − v−3z − v−3z−1 − v−5z5 + v−5z3 + v−5z + v−5z−1

S3(579) = v5zt2−1 + v4z2t2 + v2z2t2 + v2t2S3(579) = −v−5z3t3−1t1 − v−3z3t2−1 − v−3zt2−1 + 2v−5z3t2−1 + 2v−6z2t−1t1t2 − 2v−5z3t21 − v−5zt21 + v−4z2t2 +v−4t2 − v−6z2t2S3(580) = −v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 3v−3z − v−3z−1 + v−5z3 + 3v−5z + v−5z−1

S3(580) = −v−3z3t−1t1+vz3t41 +z4t21 t2−z2t21 t2−v−1z3t22−v−1zt22−v−1z3−v−1z−v−1z−1+v−3z3+v−3z+v−3z−1

S3(581) = v3zt22 − v3z − v3z−1 + vz + vz−1

S3(581) = −v−5z5t4−1 − 2v−5z3t4−1 + 3v−10z4t2−1t2 + 5v−10z2t2−1t2 − 6v−9z5t−1t1 − 12v−9z3t−1t1 − 2v−9zt−1t1 −v−7z3t22 − 2v−7zt22 − 3v−7z5 − 7v−7z3 − 3v−7z − v−7z−1 + 3v−9z5 + 7v−9z3 + 3v−9z + v−9z−1

S3(582) = vzt22 − v3z−1 + vz−1

S3(582) = −v−3z3t4−1 − v−5z3t4−1 + 2v−8z2t2−1t2 + 3v−10z2t2−1t2 − 4v−7z3t−1t1 − 6v−9z3t−1t1 − 2v−9zt−1t1 −v−5zt22 − v−7zt22 − 2v−5z3 − v−7z3 − 2v−7z − v−7z−1 + 3v−9z3 + 2v−9z + v−9z−1

S3(583) = z4t−1t21 −v−1z3t−1t2+v−1zt−1t2+v5z3t31 t2−2v4z2t1t22+v2z6t1+5v2z4t1+4v2z2t1+v2t1−z4t1−3z2t1S3(583) = −2z2t−1 + v−2t−1 + v2z2t31 + vz3t1t2 − vzt1t2S3(584) = −v−5zt2−1 − v−3z3t21 − v−3zt21 + 2v−4z2t2 + v−4t2S3(584) = v−12z4t4−1t2 − 4v−11z5t3−1t1 − 2v−11z3t3−1t1 − 2v−13z3t2−1t22 − v−9z9t2−1 − 8v−9z7t2−1 − 22v−9z5t2−1 −18v−9z3t2−1 − 6v−9zt2−1 + 6v−11z5t2−1 + 6v−11z3t2−1 + 3v−11zt2−1 + 8v−12z4t−1t1t2 + 2v−12z2t−1t1t2 + v−9z5t21 −4v−11z5t21 − 2v−11z3t21 + v−6z2t32 + v−10z8t2 + 6v−10z6t2 + 11v−10z4t2 + 8v−10z2t2 + 3v−10t2 − 4v−12z4t2 −2v−12z2t2 − 2v−12t2S3(585) = −v−5z5t6−1+5v−14z4t4−1t2−20v−13z5t3−1t1−4v−13z3t3−1t1−6v−15z3t2−1t22−5v−11z9t2−1−30v−11z7t2−1−64v−11z5t2−1−29v−11z3t2−1−6v−11zt2−1+30v−13z5t2−1+12v−13z3t2−1+3v−13zt2−1+24v−14z4t−1t1t2+6v−14z2t−1t1t2+v−11z5t21 − 12v−13z5t21 − 6v−13z3t21 − v−13zt21 + v−8z2t32 + 5v−12z8t2 + 16v−12z6t2 + 27v−12z4t2 + 14v−12z2t2 +3v−12t2 − 12v−14z4t2 − 6v−14z2t2 − 2v−14t2

Knots in L(5, 1)

S3(22) = v−2t−2 − v−1zt1t2S3(23) = −2v−3zt−2t−1 + v−2z2t3−1 + v−4t2S3(32) = −v−3zt2−2 + v−2z2t−2t2−1 − v−5zt−1t2 + v−4z2t1 + v−4t1S3(32) = −v−3zt−2t1 + v−2z2t−1 + v−2t−1S3(35) = v−2t−1 − v−1zt22S3(35) = v−2z2t−2t2−1 − 2v−5zt−1t2 + 2v−4z2t1 + v−4t1S3(36) = t−1S3(36) = −v−5zt2−2 + 3v−4z2t−2t2−1 − v−3z3t4−1 − 2v−7zt−1t2 + 2v−6z2t1 + v−6t1S3(41) = v3zt−2t−1 + v2z2t2 + v2t2S3(41) = 2v−4z2t−2t−1t1 − v−4z2t−2 + v−4t−2 − v−3z3t3−1t1 − v−3zt2−1 − v−3zt1t2


S3(42) = 2v−4z2t2−2t−1−v−3z3t−2t3−1−2v−9zt−2t2+v−8z2t2−1t2−v−7zt−1t1−v−5z3−5v−5z−v−5z−1+v−7z3+5v−7z + v−7z−1

S3(44) = vzt−2t−1 − z2t−1t1t2 − vzt21 + z2t2 + t2S3(44) = v−2z2t−2t−1t1 − v−2z2t−2 + v−2t−2 − v−1zt1t2S3(45) = −2v−7zt−2t2 + 2v−5z3t−1t1 + v−2z2t1t22 − 4v−3z − v−3z−1 + 4v−5z + v−5z−1

S3(45) = v−6z2t−2t21 − 2v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 2v−3z − v−3z−1 + v−5z3 + 2v−5z + v−5z−1

S3(48) = −v−3z3t−2t−1 − 2v−3zt−2t−1 + v−4z2t2 + v−4t2S3(48) = z2t−2 + t−2 − v2z2t31 − vz3t1t2S3(411) = t−2 − z2t31 − vzt1t2 + v−1zt1t2S3(411) = −v−1zt−2t−1 − v−3zt−2t−1 + v−4t2S3(412) = vzt−2t−1 − z2t−1t1t2 + v2t2 + z2t2S3(412) = v−2z2t−2t−1t1 − v−2z2t−2 + v−4t−2 − v−3zt2−1 − v−1zt1t2S3(414) = −v−7zt−2t2 + v−5z3t−1t1 − v−5zt−1t1 + v−2z2t1t22 − 3v−3z − v−3z−1 + 3v−5z + v−5z−1

S3(416) = v2t−2 + vzt1t2S3(416) = −v−3zt−2t−1 − 2v−5zt−2t−1 + v−2z2t3−1 + v−4z2t3−1 + v−6t2S3(417) = v−1zt−2t−1 + v−1z3t−1t31 − 2v−2z2t−1t1t2 − vzt21 + t2 + v−2z2t2S3(417) = −vzt−2t−1 − z2t2 + v−2t2S3(418) = −v−5zt−2t2 + v−3z3t−1t1 − 2v−1z − v−1z−1 + 2v−3z + v−3z−1

S3(418) = v−4z2t2−2t−1−v−3z3t−2t3−1−v−9zt−2t2+2v−8z2t2−1t2−3v−7z3t−1t1−2v−7zt−1t1−2v−5z3−4v−5z−v−5z−1 + 2v−7z3 + 4v−7z + v−7z−1

S3(421) = v−6z2t−2t21 −v−7zt−2t2−v−5z3t−1t1−v−5zt−1t1−v−3z3−3v−3z−v−3z−1+v−5z3+3v−5z+v−5z−1

S3(422) = −2v−5z3t−2t−1 − 3v−5zt−2t−1 + v−4z4t3−1 + 2v−4z2t3−1 + v−6z2t2 + v−6t2S3(424) = 3v−6z2t2−2t−1 −4v−5z3t−2t3−1 −2v−11zt−2t2 +v−4z4t5−1 + 3v−10z2t2−1t2 −4v−9z3t−1t1 −2v−9zt−1t1 −3v−7z3 − 6v−7z − v−7z−1 + 3v−9z3 + 6v−9z + v−9z−1

S3(427) = −v−2z2t−2 + v−2t−2 + z2t31 + v−1z3t1t2 − v−1zt1t2S3(52) = v3zt−2t−1 − v2z2t−1t1t2 + 2v2z2t2 + v2t2S3(52) = v−4z2t−2t−1t1 + v−4t−2 − v−3z3t2−1 − v−3zt2−1 − v−3zt1t2S3(54) = −v−5z3t2−2−v−5zt2−2+v−4z4t−2t2−1+2v−4z2t−2t2−1−v−7z3t−1t2−2v−7zt−1t2+v−6z4t1+3v−6z2t1+v−6t1S3(54) = −v−1z3t−2t1 + z4t−1 + z2t−1 + t−1 − v2z2t21 t2S3(55) = v−6z2t2−2t1−3v−5z3t−2t2−1t1+2v−5z3t−2t−1−2v−5zt−2t−1+v−4z4t4−1t1+v−4z2t3−1+2v−6z2t−1t1t2−2v−5z3t21 − v−5zt21 − v−6z2t2 + v−6t2S3(56) = v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1+4v−12z2t−2t−1t2−4v−11z3t−2t1−2v−11zt−2t1−v−11z3t3−1t2−v−10z4t2−1t1+v−10z2t2−1t1+v−8z6t−1+8v−8z4t−1+14v−8z2t−1+2v−8t−1+v−10z4t−1−8v−10z2t−1−v−10t−1−v−9zt22S3(56) = v−2z2t1 + 2v−2t1 − v−4t1S3(58) = −z2t−2t−1t1 + z2t−2 + t−2 + vz3t2−1 + vzt2−1 − vz3t1t2 − vzt1t2S3(58) = v−4z2t2−2t1−v−3z3t−2t2−1t1+v−3z3t−2t−1−v−3zt−2t−1+v−4z2t−1t1t2−v−3z3t21 −v−3zt21 −v−4z2t2+v−4t2S3(59) = v−4z2t3−2 − v−3z3t2−2t2−1 + 2v−10z2t−2t−1t2 − 2v−9z3t−2t1 − 2v−9zt−2t1 − 2v−8z4t2−1t1 + 2v−6z4t−1 +7v−6z2t−1 + v−6t−1 + 2v−8z4t−1 − 2v−8z2t−1 − v−7zt22S3(59) = v−6z2t−1t21 + v−4z4t1 + 4v−4z2t1 + 3v−4t1 − 2v−6z2t1 − 2v−6t1S3(512) = z2t−2 + t−2 + vzt2−1 − z2t−1t22 − vzt1t2S3(512) = −v−3z3t−2t2−1t1 +2v−3z3t−2t−1 −v−3zt−2t−1 +2v−4z2t−1t1t2 −2v−3z3t21 −v−3zt21 −v−4z2t2 +v−4t2S3(514) = −v−2z2t−2t−1t1+ t−2+v−2z2t−2+v−1z3t2−1+v−1zt2−1+v−1z3t−1t21 t2−v−2z2t−1t22−vzt1t2−v−1z3t1t2S3(514) = v−1z3t−2t−1 − v−3zt−2t−1 + v−2z2t−1t1t2 − v−1z3t21 − v−1zt21 − v−2z2t2 + v−4t2


S3(515) = −v−3z3t2−2t2−1+3v−10z2t−2t−1t2−3v−9z3t−2t1−v−9zt−2t1−3v−8z4t2−1t1+3v−6z4t−1+9v−6z2t−1+2v−6t−1 + 3v−8z4t−1 − 5v−8z2t−1 − v−8t−1 − v−7zt22S3(515) = v−8z2t−2t1t2 − v−6z4t−1t21 − v−7z3t−1t2 − v−7zt−1t2 + v−4z4t1 + 4v−4z2t1 + 2v−4t1 + v−6z4t1 −v−6z2t1 − v−6t1S3(516) = v−2z2t−2t−1t2 − v−1z3t−2t1 − 2z2t−1 + v−2t−1 − vzt22S3(516) = vzt2−2 − v2z2t−2t2−1 − v−2z2t−2t1t2 + v−1z3t2−1t1t2 − v−1z3t−1t2 + v2t1 + z4t1 + 2z2t1S3(517) = −v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−5zt−1t2 − v−7zt−1t2 + v−4z2t1 + v−6z2t1 + v−6t1S3(517) = −v−1zt−2t1 + v−3zt−2t1 + z2t−1 + t−1 − v−2z2t−1 − z2t21 t2S3(519) = −v−2z2t−2t−1t1 + v−2z2t−2 + v−2t−2 + v−1z3t2−1 + v−1zt2−1 + v−1z3t−1t21 t2 − v−2z2t−1t22 + z2t31 −v−1z3t1t2 − 2v−1zt1t2S3(519) = v−1z3t−2t−1 + v−2z2t−1t1t2 − v−1z3t21 − v−1zt21 − v−2z2t2 + v−2t2S3(520) = −v−5zt2−2 + v−8z2t−2t1t2 − v−6z4t−1t21 − v−7z3t−1t2 − v−7zt−1t2 + v−4z4t1 + 4v−4z2t1 + v−4t1 +v−6z4t1 − v−6z2t1S3(520) = v−10z2t−2t−1t2 − v−9z3t−2t1 − v−9zt−2t1 − v−9z3t3−1t2 + 2v−8z4t2−1t1 + 2v−8z2t2−1t1 + v−6z6t−1 +6v−6z4t−1 + 9v−6z2t−1 + 3v−6t−1 − 2v−8z4t−1 − 5v−8z2t−1 − 2v−8t−1S3(521) = −v−3z3t−2t−1t21 +v−4z2t−2t−1t2+v−3z3t−2t1−v−3zt−2t1−v−2z4t−1−2v−2z2t−1+v−4t−1+z2t21 t2−v−1zt22S3(521) = v3zt2−2 − z2t−2t1t2 − v2z2t−1t21 + vz3t−1t2 + vzt−1t2 + v4t1 + 2v2z2t1S3(522) = −vzt−2t−1 + v−1zt21 − z2t2 + t2S3(522) = −2v−2z2t−2t−1t1 + v−2z2t−2 + v−2t−2 + v−1z3t3−1t1 − vzt2−1 − v−1zt2−1 + v−1zt1t2S3(524) = v−1zt−2t1 − z2t−1 + t−1 − vzt22S3(524) = 3v−4z2t−2t2−1 − v−3z3t4−1 − 3v−7zt−1t2 + 3v−6z2t1 + v−6t1S3(525) = v5zt−2t−1 + v3z3t21 + v3zt21 + 2v4z2t2 + v4t2S3(525) = 2v−6z2t−2t−1t1 + v−6t−2 − v−5z3t3−1t1 − v−3z3t2−1 − v−3zt2−1 − v−5z3t2−1 − v−5zt2−1 − v−5zt1t2S3(528) = −v−3z3t2−2 − v−3zt2−2 − v−5zt−1t2 + 2v−4z2t1 + v−4t1S3(528) = −2v−3z3t−2t1 − v−3zt−2t1 + 2v−2z4t−1 + 2v−2z2t−1 + v−2t−1 + vz3t41 + z4t21 t2 − z2t21 t2S3(532) = t−2 − v−2z2t−2 − vzt1t2 + v−1z3t1t2 + v−1zt1t2S3(532) = v−1z3t−2t−1 − v−1zt−2t−1 − v−3zt−2t−1 + z2t3−1 − v−2z2t2 + v−4t2S3(534) = −v−3zt−2t1 + v−2t−1 + z2t21 t2 + v−1z3t22S3(534) = −v−3zt2−2 − v−2z4t−2t2−1 + 2v−2z2t−2t2−1 − v−1z3t4−1 + 2v−5z3t−1t2 − v−5zt−1t2 − 2v−4z4t1 + v−4t1S3(536) = z2t−2 + t−2 + v3zt1t2 − vz3t1t2 − vzt1t2S3(536) = −v−3z3t−2t−1 − v−3zt−2t−1 − v−5zt−2t−1 + v−4z2t3−1 + v−4z2t2 + v−4t2S3(539) = −v−3zt−2t−1 + v−2z2t2 + v−2t2S3(539) = v−2z2t−2 + v−2t−2 − v2z2t31 + z4t31 + z2t31 + vzt1t2 − 2v−1z3t1t2 − 2v−1zt1t2S3(541) = −v−5z3t2−2−2v−5zt2−2+v−4z4t−2t2−1+2v−4z2t−2t2−1−v−7z3t−1t2−v−7zt−1t2+v−6z4t1+2v−6z2t1+v−6t1S3(541) = −v−1z3t−2t1 − v−1zt−2t1 + z4t−1 + 2z2t−1 + t−1 − v2z2t21 t2 + vzt22S3(543) = −v−3zt−2t1 + v−5zt−2t1 + v−2z2t−1 + v−2t−1 − v−4z2t−1 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − v−1zt22S3(543) = −v−3zt−1t2 − v−5zt−1t2 + v−2z2t1 + v−4z2t1 + v−4t1S3(544) = −z2t−2t−1t1 + v2t−2 + z2t−2 + vz3t2−1 + vzt2−1 − vz3t1t2S3(544) = v−4z2t2−2t1 − v−3z3t−2t2−1t1 + v−3z3t−2t−1 − 2v−5zt−2t−1 + v−4z2t3−1 + v−4z2t−1t1t2 − v−3z3t21 −v−3zt21 − v−4z2t2 + v−6t2S3(546) = v3zt2−2 − z2t−2t1t2 + vz3t−1t2 + 2v2z2t1 + v2t1S3(546) = −v−3z3t−2t−1t21 + v−4z2t−2t−1t2 + v−3z3t−2t1 − v−2z4t−1 − 2v−2z2t−1 + v−2t−1 + z2t21 t2 − v−1zt22S3(547) = v−4z2t3−2 − v−3z3t2−2t2−1 + v−10z2t−2t−1t2 − v−9z3t−2t1 − 2v−9zt−2t1 − v−8z4t2−1t1 + v−8z2t2−1t1 +


2v−6z4t−1 + 7v−6z2t−1 + 2v−6t−1 + v−8z4t−1 − 2v−8z2t−1 − v−8t−1S3(547) = −v−5zt2−2 + v−8z2t−2t1t2 − v−6z4t−1t21 − v−7z3t−1t2 + v−4z4t1 + 4v−4z2t1 + 2v−4t1 + v−6z4t1 −2v−6z2t1 − v−6t1S3(548) = −v3zt2−2 − vz3t−1t2 + v−1zt−1t2 − 2v2z2t1 + t1S3(548) = 2v−3z3t−2t−1t21 − 2v−4z2t−2t−1t2 − v−1zt−2t1 − v−3z3t−2t1 − v−2z4t3−1t21 + v−3z3t3−1t2 + z2t2−1t1 +t−1 + v−2z4t−1 + 2v−2z2t−1 − z2t21 t2 + v−1zt22S3(549) = −v−3zt2−2 − v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−7zt−1t2 + v−6z2t1 + v−6t1S3(549) = −v−1zt−2t1 + z2t−1 + t−1 − z2t21 t2 + v−1zt22S3(553) = v−1zt−2t1 + t−1 − v2z2t21 t2 − vz3t22 − vzt22S3(553) = v−4z4t−2t2−1 + 2v−4z2t−2t2−1 − 2v−7z3t−1t2 − 3v−7zt−1t2 + 2v−6z4t1 + 4v−6z2t1 + v−6t1S3(555) = v−4z2t−2t−1t1 + v−2z2t−2 + v−2t−2 − v−4z2t−2 − v−3zt1t2S3(555) = v3zt−2t−1 − v2z2t−1t1t2 − vz3t21 − vzt21 + v2z2t2 + z2t2 + t2S3(556) = −v−3z3t2−2 − 2v−3zt2−2 + v−4z2t1 + v−4t1S3(556) = −2v−3z3t−2t1 − 2v−3zt−2t1 + 2v−2z4t−1 + 3v−2z2t−1 + v−2t−1 + vz3t41 + z4t21 t2 − z2t21 t2 + v−1zt22S3(559) = v−3zt−2t1 + t−1 − v−2z2t−1 − z2t21 t2 − vzt22S3(559) = v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−5zt−1t2 − 2v−7zt−1t2 + v−4z2t1 + 2v−6z2t1 + v−6t1S3(562) = −2v−2z2t−2t−1t1 + t−2 + v−2z2t−2 + 2v−1z3t2−1 + v−1zt2−1 + v−1z3t−1t21 t2 − vzt1t2 − 2v−1z3t1t2S3(562) = v−2z2t2−2t1 − v−3zt−2t−1 − v−1zt21 − v−2z2t2 + v−4t2S3(564) = vzt2−2 − v2z2t−2t2−1 − v−2z2t−2t1t2 + v−1z3t2−1t1t2 − vzt−1t2 − v−1z3t−1t2 + z4t1 + 2z2t1 + t1S3(564) = v−2z2t−2t−1t2 − v−1z3t−2t1 + v−3zt−2t1 − v−2z2t2−1t1 − 2z2t−1 + t−1 − vzt22S3(565) = −2v−7zt−1t2 + 2v−4z2t1 + v−4t1 + 2v−6z2t1 + v−2z2t32S3(565) = −v−3z3t2−2t2−1+2v−10z2t−2t−1t2−2v−9z3t−2t1−2v−8z4t2−1t1+v−8z2t2−1t1+3v−6z4t−1+9v−6z2t−1+3v−6t−1 + 2v−8z4t−1 − 6v−8z2t−1 − 2v−8t−1 − v−7zt22S3(566) = −v−1zt2−2 − v−3zt2−2 + v−4t1S3(566) = −2v−3zt−2t1 + 2v−2z2t−1 + v−2t−1 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 + v−3zt22S3(567) = v−1zt−2t1 + v2t−1 − z2t−1S3(567) = −v−7zt2−2+2v−4z2t−2t2−1+3v−6z2t−2t2−1−v−3z3t4−1−v−5z3t4−1−v−7zt−1t2−2v−9zt−1t2+v−6z2t1+2v−8z2t1 + v−8t1S3(568) = −v−4z2t2−2t1 + 3v−3z3t−2t2−1t1 − 2v−1zt−2t−1 − 2v−3z3t−2t−1 − v−2z4t4−1t1 + z2t3−1 − 2v−4z2t−1t1t2 +2v−3z3t21 + v−3zt21 + v−2t2 + v−4z2t2S3(569) = −v−5zt−1t2 + v−2z2t1 + v−2t1 + v−4z2t1S3(569) = −2v−5z3t2−2t2−1 + v−4z4t−2t4−1 + 4v−12z2t−2t−1t2 − 4v−11z3t−2t1 − v−11zt−2t1 − 2v−11z3t3−1t2 +2v−10z4t2−1t1 + 2v−10z2t2−1t1 + 2v−8z6t−1 + 12v−8z4t−1 + 16v−8z2t−1 + 3v−8t−1 − 2v−10z4t−1 − 11v−10z2t−1 −2v−10t−1 − v−9zt22S3(570) = 2v−4z2t−2t−1t2 − v−3z3t−2t1 − v−3z3t3−1t2 − 2v−2z2t−1 + v−2t−1 − v−1zt22S3(570) = v3zt2−2 − v4z2t−2t2−1 − vz3t−1t2 + v2z4t1 + 2v2z2t1 + v2t1S3(572) = v5zt2−2 + v3z3t−1t2 + v3zt−1t2 + 2v4z2t1 + v4t1S3(572) = −2v−5z3t−2t−1t21 +2v−6z2t−2t−1t2+v−5z3t−2t1−v−5zt−2t1+v−4z4t3−1t21 −v−5z3t3−1t2+v−4z2t2−1t1−v−4z4t−1 − 2v−4z2t−1 + v−4t−1 + v−2z2t21 t2 − v−3zt22S3(573) = 2v−10z2t−2t−1t2 − 2v−9z3t−2t1 − v−9zt−2t1 − v−9z3t3−1t2 + v−8z4t2−1t1 + v−8z2t2−1t1 + v−6z6t−1 +6v−6z4t−1 + 9v−6z2t−1 + 2v−6t−1 − v−8z4t−1 − 5v−8z2t−1 − v−8t−1 − v−7zt22S3(573) = v−6z2t−1t21 − v−7zt−1t2 + v−4z4t1 + 4v−4z2t1 + 2v−4t1 − v−6z2t1 − v−6t1S3(574) = −v−7z3t2−2 − v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 − 2v−5z3t4−1 − 2v−9z3t−1t2 −3v−9zt−1t2 + 2v−8z4t1 + 4v−8z2t1 + v−8t1S3(575) = vzt−2t−1 + vz3t−1t31 − 2z2t−1t1t2 − vz3t21 − vzt21 + 2z2t2 + t2


S3(576) = −2v−1zt−2t−1 + 2v−3z3t−2t−1 + v−3zt−2t−1 + z2t3−1 − v−2z4t3−1 − v−2z2t3−1 + v−2t2 − v−4z2t2S3(577) = v−3z3t−2t1 − v−2z4t−1 − v−2z2t−1 + v−2t−1 + z2t21 t2 − v−1zt22S3(577) = v−3z3t2−2−v−2z4t−2t2−1+2v−2z2t−2t2−1−v−1z3t4−1+v−5z3t−1t2−2v−5zt−1t2−v−4z4t1+v−4z2t1+v−4t1S3(579) = v5zt−2t−1 + v4z2t2 + v2z2t2 + v2t2S3(579) = 2v−6z2t−2t−1t1 + v−4z2t−2 + v−4t−2 − v−6z2t−2 − v−5z3t3−1t1 − v−3z3t2−1 − v−3zt2−1 − v−5zt1t2S3(580) = −v−5z3t−1t2 − 2v−5zt−1t2 + v−4z4t1 + 3v−4z2t1 + v−4t1S3(580) = −v−3z3t−2t1 + v−2z4t−1 + v−2z2t−1 + v−2t−1 + vz3t41 + z4t21 t2 − z2t21 t2 − v−1z3t22 − v−1zt22S3(581) = v2z2t−1 + v2t−1 + v3zt22S3(581) = −v−7z3t2−2 − 2v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 − 2v−5z3t4−1 − 2v−9z3t−1t2 −2v−9zt−1t2 + 2v−8z4t1 + 3v−8z2t1 + v−8t1S3(582) = v2t−1 + vzt22S3(582) = −v−5zt2−2−v−7zt2−2+2v−4z2t−2t2−1+3v−6z2t−2t2−1−v−3z3t4−1−v−5z3t4−1−2v−9zt−1t2+2v−8z2t1+v−8t1S3(583) = vzt2−2 + v−1z3t−2t31 − 2v−2z2t−2t1t2 + v−1z3t−1t2 + v2t1 + 2z2t1S3(583) = v−1z3t−2t1 − z4t−1 − 2z2t−1 + v−2t−1 + v2z2t21 t2 − vzt22S3(584) = −v−3zt2−2 + v−2z2t1 + v−2t1S3(584) = v−6z2t3−2−2v−5z3t2−2t2−1+v−4z4t−2t4−1+2v−12z2t−2t−1t2−2v−11z3t−2t1−2v−11zt−2t1−2v−11z3t3−1t2+4v−10z4t2−1t1+3v−10z2t2−1t1+2v−8z6t−1+11v−8z4t−1+14v−8z2t−1+3v−8t−1−4v−10z4t−1−8v−10z2t−1−2v−10t−1S3(585) = v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1+6v−14z2t−2t−1t2−6v−13z3t−2t1−2v−13zt−2t1−v−5z5t6−1−4v−13z3t3−1t2+6v−12z4t2−1t1+3v−12z2t2−1t1+4v−10z6t−1+21v−10z4t−1+23v−10z2t−1+3v−10t−1−6v−12z4t−1−16v−12z2t−1 − 2v−12t−1 − v−11zt22

Knots in L(6, 1)

S3(32) = −v−3zt2−2 + v−2z2t−2t2−1 − v−5zt−1t3 + v−4z2t2 + v−4t2S3(32) = v−2t−2 − v−1zt1t3S3(35) = v−2t−2 − v−1zt22S3(35) = v−2z2t−2t2−1 − 2v−5zt−1t3 + 2v−4z2t2 + v−4t2S3(36) = −v−5zt2−2 + 3v−4z2t−2t2−1 − v−3z3t4−1 − 2v−7zt−1t3 + 2v−6z2t2 + v−6t2S3(42) = 2v−4z2t2−2t−1 − v−3z3t−2t3−1 − 2v−9zt−2t3 + v−8z2t2−1t3 − v−7zt−1t2 + v−6z4t1 + 5v−6z2t1 + v−6t1S3(42) = −v−3zt−2t1 + v−2z2t−1 + v−2t−1S3(45) = v−4t−1 + v−2z2t1t22 − 2v−3zt2t3S3(45) = −2v−5zt−2t1 + 2v−4z2t−1 + v−4t−1 + v−2z2t21 t3S3(414) = −v−5zt−2t1 + v−4z2t−1 + v−4t−1 + v−2z2t1t22 − v−3zt2t3S3(418) = v−2t−1 − v−1zt2t3S3(418) = v−4z2t2−2t−1 − v−3z3t−2t3−1 − v−9zt−2t3 + 2v−8z2t2−1t3 − 3v−7z3t−1t2 − 2v−7zt−1t2 + 2v−6z4t1 +4v−6z2t1 + v−6t1S3(421) = −v−5zt−2t1 + v−4z2t−1 + v−4t−1 + v−2z2t21 t3 − v−3zt2t3S3(424) = t−1S3(424) = 3v−6z2t2−2t−1−4v−5z3t−2t3−1−2v−11zt−2t3+v−4z4t5−1+3v−10z2t2−1t3−4v−9z3t−1t2−2v−9zt−1t2+3v−8z4t1 + 6v−8z2t1 + v−8t1S3(54) = −v−5z3t2−2−v−5zt2−2+v−4z4t−2t2−1+2v−4z2t−2t2−1−v−7z3t−1t3−2v−7zt−1t3+v−6z4t2+3v−6z2t2+v−6t2


S3(54) = z2t−2 + t−2 − v2z2t21 t2 − vz3t1t3S3(55) = v−6z2t2−2t1−3v−5z3t−2t2−1t1+2v−5z3t−2t−1−2v−5zt−2t−1+v−4z4t4−1t1+v−4z2t3−1+2v−6z2t−1t1t3−2v−5z3t1t2 − v−5zt1t2 − v−6z2t3 + v−6t3S3(56) = v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1+4v−12z2t−2t−1t3−4v−11z3t−2t2−2v−11zt−2t2−v−11z3t3−1t3−v−10z4t2−1t2+v−10z2t2−1t2+v−9z5t−1t1−8v−9z3t−1t1−v−9zt−1t1−v−7zt23 −v−7z5−9v−7z3−5v−7z−v−7z−1+v−9z5 + 9v−9z3 + 5v−9z + v−9z−1

S3(58) = vzt−2t−1 − z2t−1t1t3 − vzt1t2 + z2t3 + t3S3(58) = v−4z2t2−2t1 − v−3z3t−2t2−1t1 + v−3z3t−2t−1 − v−3zt−2t−1 + v−4z2t−1t1t3 − v−3z3t1t2 − v−3zt1t2 −v−4z2t3 + v−4t3S3(59) = v−4z2t3−2−v−3z3t2−2t2−1+2v−10z2t−2t−1t3−2v−9z3t−2t2−2v−9zt−2t2−2v−8z4t2−1t2+2v−7z5t−1t1−2v−7z3t−1t1 − v−5zt23 − 4v−5z3 − 4v−5z − v−5z−1 + 4v−7z3 + 4v−7z + v−7z−1

S3(59) = v−6z2t−2t21 − 2v−5z3t−1t1 − 2v−5zt−1t1 − v−3z3 − 2v−3z − v−3z−1 + v−5z3 + 2v−5z + v−5z−1

S3(512) = vzt−2t−1 − z2t−1t22 − vzt1t2 + z2t3 + t3S3(512) = −v−3z3t−2t2−1t1+2v−3z3t−2t−1−v−3zt−2t−1+2v−4z2t−1t1t3−2v−3z3t1t2−v−3zt1t2−v−4z2t3+v−4t3S3(514) = v−1zt−2t−1 + v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 − vzt1t2 + t3 + v−2z2t3S3(514) = v−1z3t−2t−1 − v−3zt−2t−1 + v−2z2t−1t1t3 − v−1z3t1t2 − v−1zt1t2 − v−2z2t3 + v−4t3S3(515) = −v−3z3t2−2t2−1+3v−10z2t−2t−1t3−3v−9z3t−2t2−v−9zt−2t2−3v−8z4t2−1t2+3v−7z5t−1t1−5v−7z3t−1t1−v−7zt−1t1 − v−5zt23 − 6v−5z3 − 3v−5z − v−5z−1 + 6v−7z3 + 3v−7z + v−7z−1

S3(515) = −v−7zt−2t2 + v−5z3t−1t1 − v−5zt−1t1 + v−2z2t1t2t3 − 3v−3z − v−3z−1 + 3v−5z + v−5z−1

S3(517) = −v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−5zt−1t3 − v−7zt−1t3 + v−4z2t2 + v−6z2t2 + v−6t2S3(517) = t−2 − z2t21 t2 − vzt1t3 + v−1zt1t3S3(519) = v−1zt−2t−1 + v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 + z2t31 − 2v−1zt1t2 + v−2z2t3 + v−2t3S3(519) = v−1z3t−2t−1 + v−2z2t−1t1t3 − v−1z3t1t2 − v−1zt1t2 − v−2z2t3 + v−2t3S3(520) = −v−7zt−2t2 + v−5z3t−1t1 + v−2z2t1t2t3 − v−3zt23 − 2v−3z − v−3z−1 + 2v−5z + v−5z−1

S3(520) = v−10z2t−2t−1t3 −v−9z3t−2t2 −v−9zt−2t2 −v−9z3t3−1t3 + 2v−8z4t2−1t2 + 2v−8z2t2−1t2 − 2v−7z5t−1t1 −5v−7z3t−1t1 − 2v−7zt−1t1 − v−5z5 − 4v−5z3 − 4v−5z − v−5z−1 + v−7z5 + 4v−7z3 + 4v−7z + v−7z−1

S3(524) = t−2 + vzt1t3 − vzt22S3(524) = 3v−4z2t−2t2−1 − v−3z3t4−1 − 3v−7zt−1t3 + 3v−6z2t2 + v−6t2S3(528) = −v−3z3t2−2 − v−3zt2−2 − v−5zt−1t3 + 2v−4z2t2 + v−4t2S3(528) = v−2z2t−2 + v−2t−2 + vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − v−1zt1t3S3(534) = −v−2z2t−2 + v−2t−2 + z2t21 t2 − v−1zt1t3 + v−1z3t22S3(534) = −v−3zt2−2 − v−2z4t−2t2−1 + 2v−2z2t−2t2−1 − v−1z3t4−1 + 2v−5z3t−1t3 − v−5zt−1t3 − 2v−4z4t2 + v−4t2S3(541) = −v−5z3t2−2−2v−5zt2−2+v−4z4t−2t2−1+2v−4z2t−2t2−1−v−7z3t−1t3−v−7zt−1t3+v−6z4t2+2v−6z2t2+v−6t2S3(541) = z2t−2 + t−2 − v2z2t21 t2 − vz3t1t3 − vzt1t3 + vzt22S3(543) = v−2t−2 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − v−1zt1t3 + v−3zt1t3 − v−1zt22S3(543) = −v−3zt−1t3 − v−5zt−1t3 + v−2z2t2 + v−4z2t2 + v−4t2S3(544) = vzt−2t−1 − z2t−1t1t3 + v2t3 + z2t3S3(544) = v−4z2t2−2t1 − v−3z3t−2t2−1t1 + v−3z3t−2t−1 − 2v−5zt−2t−1 + v−4z2t3−1 + v−4z2t−1t1t3 − v−3z3t1t2 −v−3zt1t2 − v−4z2t3 + v−6t3S3(547) = v−4z2t3−2 − v−3z3t2−2t2−1 + v−10z2t−2t−1t3 − v−9z3t−2t2 − 2v−9zt−2t2 − v−8z4t2−1t2 + v−8z2t2−1t2 +v−7z5t−1t1 − 2v−7z3t−1t1 − v−7zt−1t1 − 3v−5z3 − 5v−5z − v−5z−1 + 3v−7z3 + 5v−7z + v−7z−1

S3(547) = −v−5zt−1t1 + v−2z2t1t2t3 − v−3zt23 − v−3z − v−3z−1 + v−5z + v−5z−1

S3(549) = −v−3zt2−2 − v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−7zt−1t3 + v−6z2t2 + v−6t2S3(549) = t−2 − z2t21 t2 − vzt1t3 + v−1zt22


S3(553) = z2t−2 + t−2 − v2z2t21 t2 + vzt1t3 − vz3t22 − vzt22S3(553) = v−4z4t−2t2−1 + 2v−4z2t−2t2−1 − 2v−7z3t−1t3 − 3v−7zt−1t3 + 2v−6z4t2 + 4v−6z2t2 + v−6t2S3(556) = −v−3z3t2−2 − 2v−3zt2−2 + v−4z2t2 + v−4t2S3(556) = v−2z2t−2 + v−2t−2 + vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − 2v−1zt1t3 + v−1zt22S3(559) = t−2 − z2t21 t2 + v−1zt1t3 − vzt22S3(559) = v−2z2t−2t2−1 + v−4z2t−2t2−1 − v−5zt−1t3 − 2v−7zt−1t3 + v−4z2t2 + 2v−6z2t2 + v−6t2S3(562) = v−1zt−2t−1 + v−1z3t−1t21 t2 − 2v−2z2t−1t1t3 − vzt1t2 + t3 + v−2z2t3S3(562) = v−2z2t2−2t1 − v−3zt−2t−1 − v−1zt1t2 − v−2z2t3 + v−4t3S3(565) = −2v−7zt−2t2 + 2v−5z3t−1t1 + v−2z2t32 − 4v−3z − v−3z−1 + 4v−5z + v−5z−1

S3(565) = −v−3z3t2−2t2−1+2v−10z2t−2t−1t3−2v−9z3t−2t2−2v−8z4t2−1t2+v−8z2t2−1t2+2v−7z5t−1t1−6v−7z3t−1t1−2v−7zt−1t1 − v−5zt23 − 5v−5z3 − 2v−5z − v−5z−1 + 5v−7z3 + 2v−7z + v−7z−1

S3(566) = −v−1zt2−2 − v−3zt2−2 + v−4t2S3(566) = v−2t−2 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − 2v−1zt1t3 + v−3zt22S3(567) = v2t−2 + vzt1t3S3(567) = −v−7zt2−2+2v−4z2t−2t2−1+3v−6z2t−2t2−1−v−3z3t4−1−v−5z3t4−1−v−7zt−1t3−2v−9zt−1t3+v−6z2t2+2v−8z2t2 + v−8t2S3(568) = −v−4z2t2−2t1 + 3v−3z3t−2t2−1t1 − 2v−1zt−2t−1 − 2v−3z3t−2t−1 − v−2z4t4−1t1 + z2t3−1 − 2v−4z2t−1t1t3 +2v−3z3t1t2 + v−3zt1t2 + v−2t3 + v−4z2t3S3(568) = −vzt−2t−1 − z2t3 + v−2t3S3(569) = −v−5zt−2t2 + v−3z3t−1t1 − 2v−1z − v−1z−1 + 2v−3z + v−3z−1

S3(569) = −2v−5z3t2−2t2−1 + v−4z4t−2t4−1 + 4v−12z2t−2t−1t3 − 4v−11z3t−2t2 − v−11zt−2t2 − 2v−11z3t3−1t3 +2v−10z4t2−1t2 + 2v−10z2t2−1t2 − 2v−9z5t−1t1 − 11v−9z3t−1t1 − 2v−9zt−1t1 − v−7zt23 − 2v−7z5 − 10v−7z3 −4v−7z −v−7z−1 + 2v−9z5 + 10v−9z3 + 4v−9z + v−9z−1

S3(573) = 2v−10z2t−2t−1t3 − 2v−9z3t−2t2 − v−9zt−2t2 − v−9z3t3−1t3 + v−8z4t2−1t2 + v−8z2t2−1t2 − v−7z5t−1t1 −5v−7z3t−1t1 − v−7zt−1t1 − v−5zt23 − v−5z5 − 5v−5z3 − 3v−5z − v−5z−1 + v−7z5 + 5v−7z3 + 3v−7z + v−7z−1

S3(573) = v−6z2t−2t21 −v−7zt−2t2−v−5z3t−1t1−v−5zt−1t1−v−3z3−3v−3z−v−3z−1+v−5z3+3v−5z+v−5z−1

S3(574) = −v−7z3t2−2 − v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 − 2v−5z3t4−1 − 2v−9z3t−1t3 −3v−9zt−1t3 + 2v−8z4t2 + 4v−8z2t2 + v−8t2S3(577) = −v−2z2t−2 + v−2t−2 + z2t21 t2 + v−1z3t1t3 − v−1zt22S3(577) = v−3z3t2−2 −v−2z4t−2t2−1 +2v−2z2t−2t2−1 −v−1z3t4−1 +v−5z3t−1t3 −2v−5zt−1t3 −v−4z4t2 +v−4z2t2 +v−4t2S3(580) = −v−5z3t−1t3 − 2v−5zt−1t3 + v−4z4t2 + 3v−4z2t2 + v−4t2S3(580) = v−2z2t−2 + v−2t−2 + vz3t41 + z4t21 t2 − z2t21 t2 − v−1z3t1t3 − v−1z3t22 − v−1zt22S3(581) = v2z2t−2 + v2t−2 + v3zt22S3(581) = −v−7z3t2−2 − 2v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 − 2v−5z3t4−1 − 2v−9z3t−1t3 −2v−9zt−1t3 + 2v−8z4t2 + 3v−8z2t2 + v−8t2S3(582) = v2t−2 + vzt22S3(582) = −v−5zt2−2−v−7zt2−2+2v−4z2t−2t2−1+3v−6z2t−2t2−1−v−3z3t4−1−v−5z3t4−1−2v−9zt−1t3+2v−8z2t2+v−8t2S3(584) = −v−1zt23 − v−1z−1 + v−3z−1

S3(584) = v−6z2t3−2−2v−5z3t2−2t2−1+v−4z4t−2t4−1+2v−12z2t−2t−1t3−2v−11z3t−2t2−2v−11zt−2t2−2v−11z3t3−1t3+4v−10z4t2−1t2 + 3v−10z2t2−1t2 − 4v−9z5t−1t1 − 8v−9z3t−1t1 − 2v−9zt−1t1 − 2v−7z5 − 7v−7z3 − 6v−7z − v−7z−1 +2v−9z5 + 7v−9z3 + 6v−9z + v−9z−1

S3(585) = v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1+6v−14z2t−2t−1t3−6v−13z3t−2t2−2v−13zt−2t2−v−5z5t6−1−4v−13z3t3−1t3 + 6v−12z4t2−1t2 + 3v−12z2t2−1t2 − 6v−11z5t−1t1 − 16v−11z3t−1t1 − 2v−11zt−1t1 − v−9zt23 − 4v−9z5 −


15v−9z3 − 6v−9z − v−9z−1 + 4v−11z5 + 15v−11z3 + 6v−11z + v−11z−1

Knots in L(7, 1)

S3(32) = −v−3zt−3t−1 − v−3zt2−2 + v−2z2t−2t2−1 + v−4t3S3(32) = v−2t−3 − v−1zt1t3S3(35) = v−2t−3 − v−1zt22S3(35) = −2v−3zt−3t−1 + v−2z2t−2t2−1 + v−4t3S3(36) = −2v−5zt−3t−1 − v−5zt2−2 + 3v−4z2t−2t2−1 − v−3z3t4−1 + v−6t3S3(42) = −2v−5zt−3t−2 + v−4z2t−3t2−1 + 2v−4z2t2−2t−1 − v−3z3t−2t3−1 − v−7zt−1t3 + v−6z2t2 + v−6t2S3(42) = −v−3zt−3t1 + v−2z2t−2 + v−2t−2S3(45) = v−4t−2 + v−2z2t1t22 − 2v−3zt2t3S3(45) = −2v−5zt−3t1 + 2v−4z2t−2 + v−4t−2 + v−2z2t21 t3S3(414) = −v−5zt−3t1 + v−4z2t−2 + v−4t−2 + v−2z2t1t22 − v−3zt2t3S3(418) = v−2t−2 − v−1zt2t3S3(418) = −v−5zt−3t−2 + 2v−4z2t−3t2−1 + v−4z2t2−2t−1 − v−3z3t−2t3−1 − 2v−7zt−1t3 + 2v−6z2t2 + v−6t2S3(421) = −v−5zt−3t1 + v−4z2t−2 + v−4t−2 + v−2z2t21 t3 − v−3zt2t3S3(424) = −2v−7zt−3t−2+3v−6z2t−3t2−1+3v−6z2t2−2t−1−4v−5z3t−2t3−1+v−4z4t5−1−2v−9zt−1t3+2v−8z2t2+v−8t2S3(54) = −v−5z3t−3t−1 − 2v−5zt−3t−1 − v−5z3t2−2 − v−5zt2−2 + v−4z4t−2t2−1 + 2v−4z2t−2t2−1 + v−6z2t3 + v−6t3S3(54) = z2t−3 + t−3 − v2z2t21 t2 − vz3t1t3S3(55) = v3zt−3t−1 + v2z2t3 + v2t3S3(55) = 2v−6z2t−3t−1t1 − v−6z2t−3 + v−6t−3 + v−6z2t2−2t1 − 3v−5z3t−2t2−1t1 − 2v−5zt−2t−1 + v−4z4t4−1t1 +v−4z2t3−1 − v−5zt1t3S3(56) = −v−7zt2−3+4v−6z2t−3t−2t−1−v−5z3t−3t3−1+v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−11zt−2t3+v−10z2t2−1t3 − v−9zt−1t2 + v−8z4t1 + 5v−8z2t1 + v−8t1S3(58) = vzt−3t−1 − z2t−1t1t3 − vzt1t2 + z2t3 + t3S3(58) = v−4z2t−3t−1t1 − v−4z2t−3 + v−4t−3 + v−4z2t2−2t1 − v−3z3t−2t2−1t1 − v−3zt−2t−1 − v−3zt1t3S3(59) = −v−5zt2−3+2v−4z2t−3t−2t−1+v−4z2t3−2−v−3z3t2−2t2−1−2v−9zt−2t3+2v−7z3t−1t2+4v−6z2t1+v−6t1S3(59) = v−6z2t−3t21 − 2v−5z3t−2t1 − 2v−5zt−2t1 + v−4z4t−1 + 2v−4z2t−1 + v−4t−1S3(512) = vzt−3t−1 − z2t−1t22 − vzt1t2 + z2t3 + t3S3(512) = 2v−4z2t−3t−1t1 − v−4z2t−3 + v−4t−3 − v−3z3t−2t2−1t1 − v−3zt−2t−1 − v−3zt1t3S3(514) = v−1zt−3t−1 + v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 − vzt1t2 + t3 + v−2z2t3S3(514) = v−2z2t−3t−1t1 − v−2z2t−3 + v−4t−3 − v−3zt−2t−1 − v−1zt1t3S3(515) = −v−5zt2−3 +3v−4z2t−3t−2t−1 −v−3z3t2−2t2−1 −v−9zt−2t3 +v−7z3t−1t2 −v−7zt−1t2 +3v−6z2t1 +v−6t1S3(515) = −v−7zt−3t2 + v−5z3t−2t1 − v−5zt−2t1 + 3v−4z2t−1 + v−4t−1 + v−2z2t1t2t3S3(517) = −v−3zt−3t−1 − v−5zt−3t−1 − v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 + v−6t3S3(517) = t−3 − z2t21 t2 − vzt1t3 + v−1zt1t3S3(519) = v−1zt−3t−1 + v−1z3t−1t21 t2 − v−2z2t−1t1t3 − v−2z2t−1t22 + z2t31 − 2v−1zt1t2 + v−2z2t3 + v−2t3S3(519) = v−2z2t−3t−1t1 − v−2z2t−3 + v−2t−3 − v−1zt1t3S3(520) = −v−7zt−3t2 + v−5z3t−2t1 + 2v−4z2t−1 + v−4t−1 + v−2z2t1t2t3 − v−3zt23S3(520) = v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − v−9zt−2t3 + 2v−8z2t2−1t3 − 3v−7z3t−1t2 − 2v−7zt−1t2 + 2v−6z4t1 +4v−6z2t1 + v−6t1


S3(524) = t−3 + vzt1t3 − vzt22S3(524) = −3v−5zt−3t−1 + 3v−4z2t−2t2−1 − v−3z3t4−1 + v−6t3S3(528) = −v−3zt−3t−1 − v−3z3t2−2 − v−3zt2−2 + v−4z2t3 + v−4t3S3(528) = v−2z2t−3 + v−2t−3 + vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − v−1zt1t3S3(534) = −v−2z2t−3 + v−2t−3 + z2t21 t2 − v−1zt1t3 + v−1z3t22S3(534) = 2v−3z3t−3t−1 − v−3zt−3t−1 − v−3zt2−2 − v−2z4t−2t2−1 + 2v−2z2t−2t2−1 − v−1z3t4−1 − v−4z2t3 + v−4t3S3(541) = −v−5z3t−3t−1 − v−5zt−3t−1 − v−5z3t2−2 − 2v−5zt2−2 + v−4z4t−2t2−1 + 2v−4z2t−2t2−1 + v−6z2t3 + v−6t3S3(541) = z2t−3 + t−3 − v2z2t21 t2 − vz3t1t3 − vzt1t3 + vzt22S3(543) = v−2t−3 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − v−1zt1t3 + v−3zt1t3 − v−1zt22S3(543) = −v−1zt−3t−1 − v−3zt−3t−1 + v−4t3S3(544) = vzt−3t−1 − z2t−1t1t3 + v2t3 + z2t3S3(544) = v−4z2t−3t−1t1−v−4z2t−3+v−6t−3+v−4z2t2−2t1−v−3z3t−2t2−1t1−2v−5zt−2t−1+v−4z2t3−1−v−3zt1t3S3(547) = v−4z2t−3t−2t−1+v−4z2t3−2−v−3z3t2−2t2−1−2v−9zt−2t3+v−8z2t2−1t3−v−7zt−1t2+v−6z4t1+5v−6z2t1+v−6t1S3(547) = −v−5zt−2t1 + v−4z2t−1 + v−4t−1 + v−2z2t1t2t3 − v−3zt23S3(549) = −v−5zt−3t−1 − v−3zt2−2 − v−5zt2−2 + v−2z2t−2t2−1 + v−4z2t−2t2−1 + v−6t3S3(549) = t−3 − z2t21 t2 − vzt1t3 + v−1zt22S3(553) = z2t−3 + t−3 − v2z2t21 t2 + vzt1t3 − vz3t22 − vzt22S3(553) = −2v−5z3t−3t−1 − 3v−5zt−3t−1 + v−4z4t−2t2−1 + 2v−4z2t−2t2−1 + v−6z2t3 + v−6t3S3(556) = −v−3z3t2−2 − 2v−3zt2−2 + v−4z2t3 + v−4t3S3(556) = v−2z2t−3 + v−2t−3 + vz3t41 + z4t21 t2 − z2t21 t2 − 2v−1z3t1t3 − 2v−1zt1t3 + v−1zt22S3(559) = t−3 − z2t21 t2 + v−1zt1t3 − vzt22S3(559) = −v−3zt−3t−1 − 2v−5zt−3t−1 + v−2z2t−2t2−1 + v−4z2t−2t2−1 + v−6t3S3(562) = v−1zt−3t−1 + v−1z3t−1t21 t2 − 2v−2z2t−1t1t3 − vzt1t2 + t3 + v−2z2t3S3(562) = −v−2z2t−3 + v−4t−3 + v−2z2t2−2t1 − v−3zt−2t−1 − v−1zt1t3S3(565) = −2v−7zt−3t2 + 2v−5z3t−2t1 + 4v−4z2t−1 + v−4t−1 + v−2z2t32S3(565) = −v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 + v−8z2t2−1t3 − 2v−7z3t−1t2 − 2v−7zt−1t2 + v−6z4t1 +2v−6z2t1 + v−6t1S3(566) = −v−1zt2−2 − v−3zt2−2 + v−4t3S3(566) = v−2t−3 + v−1z3t41 + z2t21 t2 − 2v−2z2t21 t2 − 2v−1zt1t3 + v−3zt22S3(567) = v2t−3 + vzt1t3S3(567) = −v−5zt−3t−1−2v−7zt−3t−1−v−7zt2−2+2v−4z2t−2t2−1+3v−6z2t−2t2−1−v−3z3t4−1−v−5z3t4−1+v−8t3S3(568) = −2v−4z2t−3t−1t1 + v−2t−3 + v−4z2t−3 − v−4z2t2−2t1 + 3v−3z3t−2t2−1t1 − 2v−1zt−2t−1 − v−2z4t4−1t1 +z2t3−1 + v−3zt1t3S3(568) = −vzt−3t−1 − z2t3 + v−2t3S3(569) = −v−5zt−3t2 + v−3z3t−2t1 + 2v−2z2t−1 + v−2t−1S3(569) = −v−7zt2−3+4v−6z2t−3t−2t−1−2v−5z3t−3t3−1−2v−5z3t2−2t2−1+v−4z4t−2t4−1−v−11zt−2t3+2v−10z2t2−1t3−3v−9z3t−1t2 − 2v−9zt−1t2 + 2v−8z4t1 + 4v−8z2t1 + v−8t1S3(573) = −v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − v−9zt−2t3 + v−8z2t2−1t3 − v−7z3t−1t2 − v−7zt−1t2 +v−6z4t1 + 3v−6z2t1 + v−6t1S3(573) = v−6z2t−3t21 − v−7zt−3t2 − v−5z3t−2t1 − v−5zt−2t1 + v−4z4t−1 + 3v−4z2t−1 + v−4t−1S3(574) = −2v−7z3t−3t−1 − 3v−7zt−3t−1 − v−7z3t2−2 − v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 −2v−5z3t4−1 + v−8z2t3 + v−8t3S3(577) = −v−2z2t−3 + v−2t−3 + z2t21 t2 + v−1z3t1t3 − v−1zt22S3(577) = v−3z3t−3t−1 − 2v−3zt−3t−1 + v−3z3t2−2 − v−2z4t−2t2−1 + 2v−2z2t−2t2−1 − v−1z3t4−1 − v−4z2t3 + v−4t3


S3(580) = −v−3z3t−3t−1 − 2v−3zt−3t−1 + v−4z2t3 + v−4t3S3(580) = v−2z2t−3 + v−2t−3 + vz3t41 + z4t21 t2 − z2t21 t2 − v−1z3t1t3 − v−1z3t22 − v−1zt22S3(581) = v2z2t−3 + v2t−3 + v3zt22S3(581) = −2v−7z3t−3t−1 − 2v−7zt−3t−1 − v−7z3t2−2 − 2v−7zt2−2 + 3v−6z4t−2t2−1 + 5v−6z2t−2t2−1 − v−5z5t4−1 −2v−5z3t4−1 + v−8z2t3 + v−8t3S3(582) = v2t−3 + vzt22S3(582) = −2v−7zt−3t−1 − v−5zt2−2 − v−7zt2−2 + 2v−4z2t−2t2−1 + 3v−6z2t−2t2−1 − v−3z3t4−1 − v−5z3t4−1 + v−8t3S3(584) = v−2t−1 − v−1zt23S3(584) = 2v−6z2t−3t−2t−1−2v−5z3t−3t3−1+v−6z2t3−2−2v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−11zt−2t3+3v−10z2t2−1t3−4v−9z3t−1t2 − 2v−9zt−1t2 + 3v−8z4t1 + 6v−8z2t1 + v−8t1S3(585) = t−1S3(585) = −v−9zt2−3+6v−8z2t−3t−2t−1−4v−7z3t−3t3−1+v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1−2v−13zt−2t3−v−5z5t6−1 + 3v−12z2t2−1t3 − 4v−11z3t−1t2 − 2v−11zt−1t2 + 3v−10z4t1 + 6v−10z2t1 + v−10t1

Knots in L(8, 1)

S3(42) = −2v−5zt−3t−2 + v−4z2t−3t2−1 + 2v−4z2t2−2t−1 − v−3z3t−2t3−1 − v−7zt−1t4 + v−6z2t3 + v−6t3S3(42) = v−2t−3 − v−1zt1t4S3(45) = v−4t−3 + v−2z2t1t22 − 2v−3zt2t3S3(45) = v−4t−3 + v−2z2t21 t3 − 2v−3zt1t4S3(414) = v−4t−3 + v−2z2t1t22 − v−3zt1t4 − v−3zt2t3S3(418) = v−2t−3 − v−1zt2t3S3(418) = −v−5zt−3t−2 + 2v−4z2t−3t2−1 + v−4z2t2−2t−1 − v−3z3t−2t3−1 − 2v−7zt−1t4 + 2v−6z2t3 + v−6t3S3(421) = v−4t−3 + v−2z2t21 t3 − v−3zt1t4 − v−3zt2t3S3(424) = −2v−7zt−3t−2+3v−6z2t−3t2−1+3v−6z2t2−2t−1−4v−5z3t−2t3−1+v−4z4t5−1−2v−9zt−1t4+2v−8z2t3+v−8t3S3(56) = −v−7zt2−3+4v−6z2t−3t−2t−1−v−5z3t−3t3−1+v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−11zt−2t4+v−10z2t2−1t4 − v−9zt−1t3 + v−8z4t2 + 5v−8z2t2 + v−8t2S3(56) = −v−3zt−3t1 + v−2z2t−2 + v−2t−2S3(59) = −v−5zt2−3+2v−4z2t−3t−2t−1+v−4z2t3−2−v−3z3t2−2t2−1−2v−9zt−2t4+2v−7z3t−1t3+4v−6z2t2+v−6t2S3(59) = −2v−5zt−3t1 + 2v−4z2t−2 + v−4t−2 + v−2z2t21 t4S3(515) = −v−5zt2−3+3v−4z2t−3t−2t−1−v−3z3t2−2t2−1−v−9zt−2t4+v−7z3t−1t3−v−7zt−1t3+3v−6z2t2+v−6t2S3(515) = −v−5zt−3t1 + v−4z2t−2 + v−4t−2 + v−2z2t1t2t3 − v−3zt2t4S3(520) = v−4t−2 + v−2z2t1t2t3 − v−3zt2t4 − v−3zt23S3(520) = v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − v−9zt−2t4 + 2v−8z2t2−1t4 − 3v−7z3t−1t3 − 2v−7zt−1t3 + 2v−6z4t2 +4v−6z2t2 + v−6t2S3(547) = v−4z2t−3t−2t−1 + v−4z2t3−2 − v−3z3t2−2t2−1 − 2v−9zt−2t4 + v−8z2t2−1t4 − v−7zt−1t3 + v−6z4t2 +5v−6z2t2 + v−6t2S3(547) = −v−5zt−3t1 + v−4z2t−2 + v−4t−2 + v−2z2t1t2t3 − v−3zt23S3(565) = v−4t−2 + v−2z2t32 − 2v−3zt2t4S3(565) = −v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 + v−8z2t2−1t4 − 2v−7z3t−1t3 − 2v−7zt−1t3 + v−6z4t2 +2v−6z2t2 + v−6t2S3(569) = v−2t−2 − v−1zt2t4


S3(569) = −v−7zt2−3+4v−6z2t−3t−2t−1−2v−5z3t−3t3−1−2v−5z3t2−2t2−1+v−4z4t−2t4−1−v−11zt−2t4+2v−10z2t2−1t4−3v−9z3t−1t3 − 2v−9zt−1t3 + 2v−8z4t2 + 4v−8z2t2 + v−8t2S3(573) = −v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − v−9zt−2t4 + v−8z2t2−1t4 − v−7z3t−1t3 − v−7zt−1t3 +v−6z4t2 + 3v−6z2t2 + v−6t2S3(573) = −v−5zt−3t1 + v−4z2t−2 + v−4t−2 + v−2z2t21 t4 − v−3zt2t4S3(584) = v−2t−2 − v−1zt23S3(584) = 2v−6z2t−3t−2t−1−2v−5z3t−3t3−1+v−6z2t3−2−2v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−11zt−2t4+3v−10z2t2−1t4−4v−9z3t−1t3 − 2v−9zt−1t3 + 3v−8z4t2 + 6v−8z2t2 + v−8t2S3(585) = −v−9zt2−3+6v−8z2t−3t−2t−1−4v−7z3t−3t3−1+v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1−2v−13zt−2t4−v−5z5t6−1 + 3v−12z2t2−1t4 − 4v−11z3t−1t3 − 2v−11zt−1t3 + 3v−10z4t2 + 6v−10z2t2 + v−10t2

Knots in L(9, 1)

S3(42) = −v−5zt−4t−1 − 2v−5zt−3t−2 + v−4z2t−3t2−1 + 2v−4z2t2−2t−1 − v−3z3t−2t3−1 + v−6t4S3(42) = v−2t−4 − v−1zt1t4S3(45) = v−4t−4 + v−2z2t1t22 − 2v−3zt2t3S3(45) = v−4t−4 + v−2z2t21 t3 − 2v−3zt1t4S3(414) = v−4t−4 + v−2z2t1t22 − v−3zt1t4 − v−3zt2t3S3(418) = v−2t−4 − v−1zt2t3S3(418) = −2v−5zt−4t−1 − v−5zt−3t−2 + 2v−4z2t−3t2−1 + v−4z2t2−2t−1 − v−3z3t−2t3−1 + v−6t4S3(421) = v−4t−4 + v−2z2t21 t3 − v−3zt1t4 − v−3zt2t3S3(424) = −2v−7zt−4t−1 − 2v−7zt−3t−2 + 3v−6z2t−3t2−1 + 3v−6z2t2−2t−1 − 4v−5z3t−2t3−1 + v−4z4t5−1 + v−8t4S3(56) = −2v−7zt−4t−2+v−6z2t−4t2−1−v−7zt2−3+4v−6z2t−3t−2t−1−v−5z3t−3t3−1+v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1 − v−9zt−1t4 + v−8z2t3 + v−8t3S3(56) = −v−3zt−4t1 + v−2z2t−3 + v−2t−3S3(59) = −2v−5zt−4t−2 − v−5zt2−3 + 2v−4z2t−3t−2t−1 + v−4z2t3−2 − v−3z3t2−2t2−1 + v−6t3S3(59) = −2v−5zt−4t1 + 2v−4z2t−3 + v−4t−3 + v−2z2t21 t4S3(515) = −v−5zt−4t−2 − v−5zt2−3 + 3v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 − v−7zt−1t4 + v−6z2t3 + v−6t3S3(515) = −v−5zt−4t1 + v−4z2t−3 + v−4t−3 + v−2z2t1t2t3 − v−3zt2t4S3(520) = v−4t−3 + v−2z2t1t2t3 − v−3zt2t4 − v−3zt23S3(520) = −v−5zt−4t−2 + 2v−4z2t−4t2−1 + v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − 2v−7zt−1t4 + 2v−6z2t3 + v−6t3S3(547) = −2v−5zt−4t−2+v−4z2t−4t2−1+v−4z2t−3t−2t−1+v−4z2t3−2−v−3z3t2−2t2−1−v−7zt−1t4+v−6z2t3+v−6t3S3(547) = −v−5zt−4t1 + v−4z2t−3 + v−4t−3 + v−2z2t1t2t3 − v−3zt23S3(565) = v−4t−3 + v−2z2t32 − 2v−3zt2t4S3(565) = v−4z2t−4t2−1 − v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 − 2v−7zt−1t4 + 2v−6z2t3 + v−6t3S3(569) = v−2t−3 − v−1zt2t4S3(569) = −v−7zt−4t−2+2v−6z2t−4t2−1−v−7zt2−3+4v−6z2t−3t−2t−1−2v−5z3t−3t3−1−2v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−9zt−1t4 + 2v−8z2t3 + v−8t3S3(573) = −v−5zt−4t−2+v−4z2t−4t2−1−v−5zt2−3+2v−4z2t−3t−2t−1−v−3z3t−3t3−1−v−7zt−1t4+v−6z2t3+v−6t3S3(573) = −v−5zt−4t1 + v−4z2t−3 + v−4t−3 + v−2z2t21 t4 − v−3zt2t4S3(584) = v−2t−3 − v−1zt23S3(584) = −2v−7zt−4t−2 + 3v−6z2t−4t2−1 + 2v−6z2t−3t−2t−1 − 2v−5z3t−3t3−1 + v−6z2t3−2 − 2v−5z3t2−2t2−1 +v−4z4t−2t4−1 − 2v−9zt−1t4 + 2v−8z2t3 + v−8t3

124 Knots in L(10, 1)

S3(585) = −2v−9zt−4t−2+3v−8z2t−4t2−1−v−9zt2−3+6v−8z2t−3t−2t−1−4v−7z3t−3t3−1+v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1 − v−5z5t6−1 − 2v−11zt−1t4 + 2v−10z2t3 + v−10t3

Knots in L(10, 1)

S3(56) = −2v−7zt−4t−2+v−6z2t−4t2−1−v−7zt2−3+4v−6z2t−3t−2t−1−v−5z3t−3t3−1+v−6z2t3−2−3v−5z3t2−2t2−1+v−4z4t−2t4−1 − v−9zt−1t5 + v−8z2t4 + v−8t4S3(56) = v−2t−4 − v−1zt1t5S3(59) = −2v−5zt−4t−2 − v−5zt2−3 + 2v−4z2t−3t−2t−1 + v−4z2t3−2 − v−3z3t2−2t2−1 + v−6t4S3(59) = v−4t−4 + v−2z2t21 t4 − 2v−3zt1t5S3(515) = −v−5zt−4t−2 − v−5zt2−3 + 3v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 − v−7zt−1t5 + v−6z2t4 + v−6t4S3(515) = v−4t−4 + v−2z2t1t2t3 − v−3zt1t5 − v−3zt2t4S3(520) = v−4t−4 + v−2z2t1t2t3 − v−3zt2t4 − v−3zt23S3(520) = −v−5zt−4t−2 + 2v−4z2t−4t2−1 + v−4z2t−3t−2t−1 − v−3z3t−3t3−1 − 2v−7zt−1t5 + 2v−6z2t4 + v−6t4S3(547) = −2v−5zt−4t−2+v−4z2t−4t2−1+v−4z2t−3t−2t−1+v−4z2t3−2−v−3z3t2−2t2−1−v−7zt−1t5+v−6z2t4+v−6t4S3(547) = v−4t−4 + v−2z2t1t2t3 − v−3zt1t5 − v−3zt23S3(565) = v−4t−4 + v−2z2t32 − 2v−3zt2t4S3(565) = v−4z2t−4t2−1 − v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 − 2v−7zt−1t5 + 2v−6z2t4 + v−6t4S3(569) = v−2t−4 − v−1zt2t4S3(569) = −v−7zt−4t−2+2v−6z2t−4t2−1−v−7zt2−3+4v−6z2t−3t−2t−1−2v−5z3t−3t3−1−2v−5z3t2−2t2−1+v−4z4t−2t4−1−2v−9zt−1t5 + 2v−8z2t4 + v−8t4S3(573) = −v−5zt−4t−2+v−4z2t−4t2−1−v−5zt2−3+2v−4z2t−3t−2t−1−v−3z3t−3t3−1−v−7zt−1t5+v−6z2t4+v−6t4S3(573) = v−4t−4 + v−2z2t21 t4 − v−3zt1t5 − v−3zt2t4S3(584) = v−2t−4 − v−1zt23S3(584) = −2v−7zt−4t−2 + 3v−6z2t−4t2−1 + 2v−6z2t−3t−2t−1 − 2v−5z3t−3t3−1 + v−6z2t3−2 − 2v−5z3t2−2t2−1 +v−4z4t−2t4−1 − 2v−9zt−1t5 + 2v−8z2t4 + v−8t4S3(585) = −2v−9zt−4t−2+3v−8z2t−4t2−1−v−9zt2−3+6v−8z2t−3t−2t−1−4v−7z3t−3t3−1+v−8z2t3−2−6v−7z3t2−2t2−1+5v−6z4t−2t4−1 − v−5z5t6−1 − 2v−11zt−1t5 + 2v−10z2t4 + v−10t4

Knots in L(11, 1)

S3(56) = −v−7zt−5t−1 − 2v−7zt−4t−2 + v−6z2t−4t2−1 − v−7zt2−3 + 4v−6z2t−3t−2t−1 − v−5z3t−3t3−1 + v−6z2t3−2 −3v−5z3t2−2t2−1 + v−4z4t−2t4−1 + v−8t5S3(56) = v−2t−5 − v−1zt1t5S3(59) = −2v−5zt−4t−2 − v−5zt2−3 + 2v−4z2t−3t−2t−1 + v−4z2t3−2 − v−3z3t2−2t2−1 + v−6t5S3(59) = v−4t−5 + v−2z2t21 t4 − 2v−3zt1t5S3(515) = −v−5zt−5t−1 − v−5zt−4t−2 − v−5zt2−3 + 3v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 + v−6t5S3(515) = v−4t−5 + v−2z2t1t2t3 − v−3zt1t5 − v−3zt2t4S3(520) = v−4t−5 + v−2z2t1t2t3 − v−3zt2t4 − v−3zt23S3(520) = −2v−5zt−5t−1 − v−5zt−4t−2 + 2v−4z2t−4t2−1 + v−4z2t−3t−2t−1 − v−3z3t−3t3−1 + v−6t5


S3(547) = −v−5zt−5t−1 − 2v−5zt−4t−2 + v−4z2t−4t2−1 + v−4z2t−3t−2t−1 + v−4z2t3−2 − v−3z3t2−2t2−1 + v−6t5S3(547) = v−4t−5 + v−2z2t1t2t3 − v−3zt1t5 − v−3zt23S3(565) = v−4t−5 + v−2z2t32 − 2v−3zt2t4S3(565) = −2v−5zt−5t−1 + v−4z2t−4t2−1 − v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t2−2t2−1 + v−6t5S3(569) = v−2t−5 − v−1zt2t4S3(569) = −2v−7zt−5t−1−v−7zt−4t−2+2v−6z2t−4t2−1−v−7zt2−3+4v−6z2t−3t−2t−1−2v−5z3t−3t3−1−2v−5z3t2−2t2−1+v−4z4t−2t4−1 + v−8t5S3(573) = −v−5zt−5t−1 − v−5zt−4t−2 + v−4z2t−4t2−1 − v−5zt2−3 + 2v−4z2t−3t−2t−1 − v−3z3t−3t3−1 + v−6t5S3(573) = v−4t−5 + v−2z2t21 t4 − v−3zt1t5 − v−3zt2t4S3(584) = v−2t−5 − v−1zt23S3(584) = −2v−7zt−5t−1 − 2v−7zt−4t−2 + 3v−6z2t−4t2−1 + 2v−6z2t−3t−2t−1 − 2v−5z3t−3t3−1 + v−6z2t3−2 −2v−5z3t2−2t2−1 + v−4z4t−2t4−1 + v−8t5S3(585) = −2v−9zt−5t−1−2v−9zt−4t−2+3v−8z2t−4t2−1−v−9zt2−3+6v−8z2t−3t−2t−1−4v−7z3t−3t3−1+v−8z2t3−2−6v−7z3t2−2t2−1 + 5v−6z4t−2t4−1 − v−5z5t6−1 + v−10t5

DThe Kauffman bracket skein modules

�e Kau�man bracket skein modules of the non-a�ne prime knots are expressed in the stan-

dard basis using the evaluation x0 = −A2 − A−2 (or equivalently [∅] = 1), see�eorem 2.3.4and�eorem 2.3.5. For knots in L(p, q) that are not listed below (and all knots in L(12, 1) andL(12, 5)) the KBSM expressions equal the KBSM expressions of the corresponding knots in thesolid torus (up to a change of framing).

Knots in the solid torus

S2,∞(01) = xS2,∞(11) = −A−2x2 + A−2 + A−6

S2,∞(11) = A6 − A2x2 + A2

S2,∞(21) = −A−2x2 + A−6x2 − A−6 − A−10

S2,∞(21) = −A10 + A6x2 − A6 − A2x2

S2,∞(22) = −A4x + x3 − x − A−4xS2,∞(23) = A−4x3 − 2A−4x − A−8xS2,∞(23) = −A8x + A4x3 − 2A4xS2,∞(31) = A−4x3 − A−4x − A−8x3 + A−8x + A−12xS2,∞(31) = A12x − A8x3 + A8x + A4x3 − A4xS2,∞(32) = A6x2 − A2x4 + 2A2x2 − A2 + A−2x2 − A−2

S2,∞(32) = A2x2 − A2 − A−2x4 + 2A−2x2 − A−2 + A−6x2

S2,∞(33) = −A−2x2 + A−6x2 − A−10x2 + A−10 + A−14

S2,∞(33) = A14 − A10x2 + A10 + A6x2 − A2x2

S2,∞(34) = −A4x + x3 − A−4x3 + A−4x + A−8xS2,∞(34) = A8x − A4x3 + A4x + x3 − A−4xS2,∞(35) = A2x2 − A−2x4 + A−2x2 + 2A−6x2 − A−6 − A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − A2x4 + A2x2 + A−2x2

S2,∞(36) = −A−6x4 + 3A−6x2 − A−6 + A−10x2 − A−10

S2,∞(36) = A10x2 − A10 − A6x4 + 3A6x2 − A6

S2,∞(37) = −A−6x2 + A−6 + A−18

S2,∞(37) = A18 − A6x2 + A6

S2,∞(38) = −A−2 − A−10x2 + A−14x2 − A−14

127


S2,∞(38) = A14x2 − A14 − A10x2 − A2

S2,∞(41) = −A−6x4 + 2A−6x2 + A−10x4 − 2A−10x2 + A−10 − A−14x2 + A−14

S2,∞(41) = −A14x2 + A14 + A10x4 − 2A10x2 + A10 − A6x4 + 2A6x2

S2,∞(42) = −A8x3 + A8x + A4x5 − 3A4x3 + 2A4x − x3 + 2xS2,∞(42) = −x3 + 2x + A−4x5 − 3A−4x3 + 2A−4x − A−8x3 + A−8xS2,∞(43) = A−4x3 − 2A−8x3 + 2A−8x + A−12x3 − A−16xS2,∞(43) = −A16x + A12x3 − 2A8x3 + 2A8x + A4x3

S2,∞(44) = A2x2 − A−2x4 + A−2 + A−6x4 − A−6x2 + A−6 − A−10x2

S2,∞(44) = −A10x2 + A6x4 − A6x2 + A6 − A2x4 + A2 + A−2x2

S2,∞(45) = −A4x3 + x5 − 2x3 + 2x − 2A−4x3 + 2A−4x + A−8xS2,∞(45) = A8x − 2A4x3 + 2A4x + x5 − 2x3 + 2x − A−4x3

S2,∞(46) = −A−6x2 + A−10x2 − A−10 − A−14x2 + A−18x2 − A−22

S2,∞(46) = −A22 + A18x2 − A14x2 + A10x2 − A10 − A6x2

S2,∞(47) = A−4x − A−8x + A−12x3 − A−16x3 + A−20xS2,∞(47) = A20x − A16x3 + A12x3 − A8x + A4xS2,∞(48) = −A16x + A12x − A8x + A4x3 − A4x − xS2,∞(48) = −x + A−4x3 − A−4x − A−8x + A−12x − A−16xS2,∞(49) = A14x2 − 2A10x2 + A10 + A6x2 − A2x2 + A−2

S2,∞(49) = A2 − A−2x2 + A−6x2 − 2A−10x2 + A−10 + A−14x2

S2,∞(410) = −A−2x2 + A−6x2 − A−10x2 + A−14x2 − A−14 − A−18

S2,∞(410) = −A18 + A14x2 − A14 − A10x2 + A6x2 − A2x2

S2,∞(411) = −A4x + x3 − A−4x3 + A−8x3 − A−8x − A−12xS2,∞(411) = −A12x + A8x3 − A8x − A4x3 + x3 − A−4xS2,∞(412) = A2x2 − A−2x4 + A−6x4 − 2A−10x2 + A−10 + A−14

S2,∞(412) = A14 − 2A10x2 + A10 + A6x4 − A2x4 + A−2x2

S2,∞(413) = A8x − A4x3 + 2x3 − x − A−4x3 + A−8xS2,∞(414) = A8x − 2A4x3 + A4x + x5 − x3 + x − 2A−4x3 + A−4x + A−8xS2,∞(415) = −A6x2 + A2x4 − A2x2 − A−2x4 + 2A−6x2 − A−6 − A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − A2x4 + A−2x4 − A−2x2 − A−6x2

S2,∞(416) = A−4x3 − A−4x − A−8x3 + A−12x3 − A−12x − A−16xS2,∞(416) = −A16x + A12x3 − A12x − A8x3 + A4x3 − A4xS2,∞(417) = A6x2 − A2x4 + A2x2 + A−2x4 − 2A−2x2 + A−2 − A−6x2 + A−6

S2,∞(417) = −A6x2 + A6 + A2x4 − 2A2x2 + A2 − A−2x4 + A−2x2 + A−6x2

S2,∞(418) = −x3 + x + A−4x5 − 2A−4x3 + A−4x − 2A−8x3 + 2A−8x + A−12xS2,∞(418) = A12x − 2A8x3 + 2A8x + A4x5 − 2A4x3 + A4x − x3 + xS2,∞(419) = −A10x2 + A6x4 − A6x2 − A2x4 + A2x2 − A2 + A−2x2 − A−2

S2,∞(419) = A2x2 − A2 − A−2x4 + A−2x2 − A−2 + A−6x4 − A−6x2 − A−10x2

S2,∞(420) = −A−6x4 + 2A−6x2 − A−6 + A−10x4 − A−10x2 − A−10 − A−14x2

S2,∞(420) = −A14x2 + A10x4 − A10x2 − A10 − A6x4 + 2A6x2 − A6

S2,∞(421) = −A4x3 + A4x + x5 − 3x3 + 3x − A−4x3 + A−4xS2,∞(422) = A−8x3 − 2A−8x − A−16xS2,∞(422) = −A16x + A8x3 − 2A8xS2,∞(423) = −A12x3 + 2A12x + A8x3 − A8xS2,∞(423) = A−8x3 − A−8x − A−12x3 + 2A−12xS2,∞(424) = A−8x5 − 4A−8x3 + 3A−8x − A−12x3 + 2A−12x

Appendix D. �e Kau�man bracket skein modules 129

S2,∞(424) = −A12x3 + 2A12x + A8x5 − 4A8x3 + 3A8xS2,∞(425) = −A6x2 + A6 + A2x2 − A2 − A−2 − A−10

S2,∞(425) = −A10 − A2 + A−2x2 − A−2 − A−6x2 + A−6

S2,∞(426) = −A6x2 + A6 + A2x2 − A−2x2 + A−2 + A−6 − A−10

S2,∞(426) = −A10 + A6 − A2x2 + A2 + A−2x2 − A−6x2 + A−6

S2,∞(427) = A8x − 2A4x + x3 − x − 2A−4x + A−8xS2,∞(51) = A−8x3 − A−8x − A−12x3 + 2A−12xS2,∞(51) = −A12x3 + 2A12x + A8x3 − A8xS2,∞(52) = A−6 − A−10x4 + A−10x2 + A−14x4 − A−14x2 − A−18x2 + A−18

S2,∞(52) = −A18x2 + A18 + A14x4 − A14x2 − A10x4 + A10x2 + A6

S2,∞(53) = −A12x3 + A12x + 2A8x3 − 2A8x − A4x3 + A4x + xS2,∞(53) = x − A−4x3 + A−4x + 2A−8x3 − 2A−8x − A−12x3 + A−12xS2,∞(54) = A14x2 − A14 − A6x4 + 2A6x2 + A2x2 − A2

S2,∞(54) = A−2x2 − A−2 − A−6x4 + 2A−6x2 + A−14x2 − A−14

S2,∞(55) = A−8x5 − 3A−8x3 + A−8x − A−12x5 + 3A−12x3 − 2A−12x + A−16x3 − 2A−16xS2,∞(55) = A16x3 − 2A16x − A12x5 + 3A12x3 − 2A12x + A8x5 − 3A8x3 + A8xS2,∞(56) = A10x4 − 2A10x2 − A6x6 + 4A6x4 − 4A6x2 + A6 + A2x4 − 3A2x2 + A2

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x6 + 4A−6x4 − 4A−6x2 + A−6 + A−10x4 − 2A−10x2

S2,∞(57) = −A−6x4 + A−6x2 + 2A−10x4 − 3A−10x2 − A−14x4 + A−14x2 − A−14 + A−18x2 − A−18

S2,∞(57) = A18x2 − A18 − A14x4 + A14x2 − A14 + 2A10x4 − 3A10x2 − A6x4 + A6x2

S2,∞(58) = −x3 + x + A−4x5 − A−4x3 − A−4x − A−8x5 + 2A−8x3 − 2A−8x + A−12x3 − A−12xS2,∞(58) = A12x3 − A12x − A8x5 + 2A8x3 − 2A8x + A4x5 − A4x3 − A4x − x3 + xS2,∞(59) = A6x4 − A6x2 − A2x6 + 3A2x4 − 3A2x2 + 2A−2x4 − 4A−2x2 + A−2 − A−6x2 + A−6

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − A−2x6 + 3A−2x4 − 3A−2x2 + A−6x4 − A−6x2

S2,∞(510) = A−4x3 − 2A−8x3 + A−8x + 2A−12x3 − A−12x − A−16x3 + A−20xS2,∞(510) = A20x − A16x3 + 2A12x3 − A12x − 2A8x3 + A8x + A4x3

S2,∞(511) = A2x2 − A−2x4 − A−2x2 + A−2 + 2A−6x4 − 2A−6x2 + A−6 − A−10x4 + A−14x2

S2,∞(511) = A14x2 − A10x4 + 2A6x4 − 2A6x2 + A6 − A2x4 − A2x2 + A2 + A−2x2

S2,∞(512) = −x3 + A−4x5 − A−4x − A−8x5 + 2A−12x3 − A−12x − A−16xS2,∞(512) = −A16x + 2A12x3 − A12x − A8x5 + A4x5 − A4x − x3

S2,∞(513) = −A6x2 + A2x4 − A2 − 2A−2x4 + 2A−2x2 − A−2 + A−6x4 − A−10x2

S2,∞(513) = −A10x2 + A6x4 − 2A2x4 + 2A2x2 − A2 + A−2x4 − A−2 − A−6x2

S2,∞(514) = −A4x3 + x5 − x3 + x − A−4x5 + A−4x3 − A−4x + 2A−8x3 − 2A−8x − A−12xS2,∞(514) = −A12x + 2A8x3 − 2A8x − A4x5 + A4x3 − A4x + x5 − x3 + x − A−4x3

S2,∞(515) = −A10x2 + 2A6x4 − 2A6x2 + A6 − A2x6 + 2A2x4 − 3A2x2 + A2 + 2A−2x4 − 2A−2x2 − A−6x2

S2,∞(515) = −A6x2+2A2x4−2A2x2−A−2x6+2A−2x4−3A−2x2+A−2+2A−6x4−2A−6x2+A−6−A−10x2

S2,∞(516) = A8x3 − A4x5 + A4x3 − A4x + x5 − 2A−4x3 + A−4x + A−8xS2,∞(516) = A8x − 2A4x3 + A4x + x5 − A−4x5 + A−4x3 − A−4x + A−8x3

S2,∞(517) = A14x2 − A10x4 + A10x2 − A10 + A6x4 + A6x2 − A6 − A2x4 + A−2x2

S2,∞(517) = A2x2 − A−2x4 + A−6x4 + A−6x2 − A−6 − A−10x4 + A−10x2 − A−10 + A−14x2

S2,∞(518) = −A−6x4 + A−6x2 + 2A−10x4 − 3A−10x2 + A−10 − A−14x4 + A−14 + A−18x2

S2,∞(518) = A18x2 − A14x4 + A14 + 2A10x4 − 3A10x2 + A10 − A6x4 + A6x2

S2,∞(519) = −A4x3 + x5 − x3 − A−4x5 + 2A−4x3 − 2A−4x + A−8x3 − A−8xS2,∞(519) = A8x3 − A8x − A4x5 + 2A4x3 − 2A4x + x5 − x3 − A−4x3

S2,∞(520) = A2x4 − A2x2 − A−2x6 + 3A−2x4 − 4A−2x2 + A−2 + 2A−6x4 − 3A−6x2 + A−6 − A−10x2


S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 + A6 − A2x6 + 3A2x4 − 4A2x2 + A2 + A−2x4 − A−2x2

S2,∞(521) = A12x3 − A8x5 + A8x3 + A4x5 − A4x3 − x3 + xS2,∞(521) = −x3 + x + A−4x5 − A−4x3 − A−8x5 + A−8x3 + A−12x3

S2,∞(522) = −A6x2 + A2x4 − A2x2 − A−2x4 + A−2 + A−6x2 + A−6

S2,∞(522) = A6x2 + A6 − A2x4 + A2 + A−2x4 − A−2x2 − A−6x2

S2,∞(523) = −A12x + A8x3 − A8x − A4x3 + A4x + 2xS2,∞(523) = 2x − A−4x3 + A−4x + A−8x3 − A−8x − A−12xS2,∞(524) = −A−6x4 + 3A−6x2 + A−14x2 − A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 − A6x4 + 3A6x2

S2,∞(525) = −A−6x2 + 2A−10x2 − A−14x4 − A−14x2 + A−14 + A−18x4 + A−18x2 − 2A−22x2 + A−26

S2,∞(525) = A26 − 2A22x2 + A18x4 + A18x2 − A14x4 − A14x2 + A14 + 2A10x2 − A6x2

S2,∞(526) = −x + A−4x3 + A−4x − 2A−8x3 + A−8x + 2A−12x3 − A−16x3 − A−16x + A−20xS2,∞(526) = A20x − A16x3 − A16x + 2A12x3 − 2A8x3 + A8x + A4x3 + A4x − xS2,∞(527) = −x + A−4x3 + A−4x − 2A−8x3 + A−8x + 2A−12x3 − A−16x3 − A−16x + A−20xS2,∞(527) = A20x − A16x3 − A16x + 2A12x3 − 2A8x3 + A8x + A4x3 + A4x − xS2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − A2x4 + 2A−2x2 − A−6

S2,∞(528) = −A6 + 2A2x2 − A−2x4 + 3A−6x2 − A−6 − 2A−10x2 + A−14x2

S2,∞(529) = −A6x2 + 2A2x2 − A2 − 2A−2x2 + 2A−6x2 − A−6 − A−10x2 − A−10 + A−14

S2,∞(529) = A14 − A10x2 − A10 + 2A6x2 − A6 − 2A2x2 + 2A−2x2 − A−2 − A−6x2

S2,∞(530) = A8x − A4x + 2x − A−4x3 + A−8x3 − A−12xS2,∞(530) = −A12x + A8x3 − A4x3 + 2x − A−4x + A−8xS2,∞(531) = A8x − 2A4x + x3 + x − 2A−4x3 + A−4x + A−8x3 + A−8x − A−12xS2,∞(531) = −A12x + A8x3 + A8x − 2A4x3 + A4x + x3 + x − 2A−4x + A−8xS2,∞(532) = A8x − 2A4x + x3 − A−4x3 − A−4x + A−8x3 − A−12xS2,∞(532) = −A12x + A8x3 − A4x3 − A4x + x3 − 2A−4x + A−8xS2,∞(533) = −A6x2 + 2A2x2 − A2 − 2A−2x2 + A−2 + A−6x2 + A−6 − A−10x2 + A−14

S2,∞(533) = A14 − A10x2 + A6x2 + A6 − 2A2x2 + A2 + 2A−2x2 − A−2 − A−6x2

S2,∞(534) = −A6x2 + 3A2x2 − A2 − A−2x4 + 3A−6x2 − A−6 − A−10x2 − A−10 + A−14

S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − A2x4 + 3A−2x2 − A−2 − A−6x2

S2,∞(535) = −A−6x2 + 2A−10x2 − A−10 − A−14x4 − A−14 + A−18x4 + A−18x2 − A−18 − 2A−22x2 + A−26

S2,∞(535) = A26 − 2A22x2 + A18x4 + A18x2 − A18 − A14x4 − A14 + 2A10x2 − A10 − A6x2

S2,∞(536) = −x + A−4x3 − A−8x3 − A−8x + 2A−12x3 − A−12x − A−16x3 − A−16x + A−20xS2,∞(536) = A20x − A16x3 − A16x + 2A12x3 − A12x − A8x3 − A8x + A4x3 − xS2,∞(537) = A14x2 − 2A10x2 + 2A6x2 − A6 − 2A2x2 + A−2x2 − A−6

S2,∞(537) = −A6 + A2x2 − 2A−2x2 + 2A−6x2 − A−6 − 2A−10x2 + A−14x2

S2,∞(538) = −A−2 + A−6x2 − A−6 − A−10x4 + A−10x2 − A−10 + A−14x4 − A−14x2 − A−18x2 + A−18

S2,∞(538) = −A18x2 + A18 + A14x4 − A14x2 − A10x4 + A10x2 − A10 + A6x2 − A6 − A2

S2,∞(539) = −A12x3 + A12x + 2A8x3 − 2A8x − A4x3 + x3 − x − A−4xS2,∞(539) = −A4x + x3 − x − A−4x3 + 2A−8x3 − 2A−8x − A−12x3 + A−12xS2,∞(540) = A−8x3 − A−8x − A−12x3 + A−12x + A−16x3 − A−16x − A−20x3 + A−20x + A−24xS2,∞(540) = A24x − A20x3 + A20x + A16x3 − A16x − A12x3 + A12x + A8x3 − A8xS2,∞(541) = A18 + A10x2 − A10 − A6x4 + 2A6x2 − A6 + A2x2 − A2

S2,∞(541) = A−2x2 − A−2 − A−6x4 + 2A−6x2 − A−6 + A−10x2 − A−10 + A−18

S2,∞(542) = A−4x3 − A−4x − A−8x3 + A−12x3 − A−16x3 + A−16x + A−20xS2,∞(542) = A20x − A16x3 + A16x + A12x3 − A8x3 + A4x3 − A4x


S2,∞(543) = A6x2 − A2x4 + A2x2 + A−2x4 − A−2x2 − A−6x4 + 2A−6x2 − A−6 + A−10x2 − A−10

S2,∞(543) = A10x2 − A10 − A6x4 + 2A6x2 − A6 + A2x4 − A2x2 − A−2x4 + A−2x2 + A−6x2

S2,∞(544) = −x3 + x + A−4x5 − A−4x3 − A−8x5 + A−8x3 − A−8x + 2A−12x3 − 2A−12x − A−16xS2,∞(544) = −A16x + 2A12x3 − 2A12x − A8x5 + A8x3 − A8x + A4x5 − A4x3 − x3 + xS2,∞(545) = −A10x2 + A6x4 − 2A2x4 + 2A2x2 + A−2x4 − A−2x2 + A−2 − A−6x2 + A−6

S2,∞(545) = −A6x2 + A6 + A2x4 − A2x2 + A2 − 2A−2x4 + 2A−2x2 + A−6x4 − A−10x2

S2,∞(546) = −x3 + x + A−4x5 − A−4x3 + A−4x − A−8x5 + A−8x + 2A−12x3 − A−12x − A−16xS2,∞(546) = −A16x + 2A12x3 − A12x − A8x5 + A8x + A4x5 − A4x3 + A4x − x3 + xS2,∞(547) = −A10x2 + 2A6x4 − 2A6x2 − A2x6 + 2A2x4 − 2A2x2 + 2A−2x4 − 3A−2x2 + A−2 − A−6x2 + A−6

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − A−2x6 + 2A−2x4 − 2A−2x2 + 2A−6x4 − 2A−6x2 − A−10x2

S2,∞(548) = A4x3 − A4x − x5 + 2x3 − x + A−4x5 − A−4x3 − 2A−8x3 + 2A−8x + A−12xS2,∞(548) = A12x − 2A8x3 + 2A8x + A4x5 − A4x3 − x5 + 2x3 − x + A−4x3 − A−4xS2,∞(549) = A14x2 − A10x4 + A10x2 + A6x4 − A2x4 + A2x2 − A2 + A−2x2 − A−2

S2,∞(549) = A2x2 − A2 − A−2x4 + A−2x2 − A−2 + A−6x4 − A−10x4 + A−10x2 + A−14x2

S2,∞(550) = −A−6x2 + A−10x2 − 2A−14x2 + A−14 + 2A−18x2 − A−22x2 + A−26

S2,∞(550) = A26 − A22x2 + 2A18x2 − 2A14x2 + A14 + A10x2 − A6x2

S2,∞(551) = A−4x − A−8x + A−12x3 + A−12x − 2A−16x3 + A−16x + A−20x3 − A−24xS2,∞(551) = −A24x + A20x3 − 2A16x3 + A16x + A12x3 + A12x − A8x + A4xS2,∞(552) = −A16x + A12x − A8x + A4x3 − x3 + x + A−4xS2,∞(552) = A4x − x3 + x + A−4x3 − A−8x + A−12x − A−16xS2,∞(553) = A−2x2 − A−6x4 + A−6x2 + 2A−10x2 − A−10 − A−14x2 + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − A6x4 + A6x2 + A2x2

S2,∞(554) = A20x − A16x3 + A12x3 + A12x − A8x3 + A4x3 − xS2,∞(554) = −x + A−4x3 − A−8x3 + A−12x3 + A−12x − A−16x3 + A−20xS2,∞(555) = −A18x2 + A14x4 − A10x4 − A10x2 + A10 + 2A6x2 − A2x2 + A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − A−10x4 − A−10x2 + A−10 + A−14x4 − A−18x2

S2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − A2x4 + A2x2 − A2 + 2A−2x2 − A−2 − A−6

S2,∞(556) = −A6 + 2A2x2 − A2 − A−2x4 + A−2x2 − A−2 + 2A−6x2 − 2A−10x2 + A−10 + A−14x2

S2,∞(557) = −A−2x2 + A−6x2 − A−10x2 + A−14x2 − A−18x2 + A−18 + A−22

S2,∞(557) = A22 − A18x2 + A18 + A14x2 − A10x2 + A6x2 − A2x2

S2,∞(558) = −A4x + x3 − A−4x3 + A−8x3 − A−12x3 + A−12x + A−16xS2,∞(558) = A16x − A12x3 + A12x + A8x3 − A4x3 + x3 − A−4xS2,∞(559) = A2x2 − A−2x4 + A−6x4 + A−6x2 − A−10x4 + 2A−14x2 − A−14 − A−18

S2,∞(559) = −A18 + 2A14x2 − A14 − A10x4 + A6x4 + A6x2 − A2x4 + A−2x2

S2,∞(560) = A8x − A4x3 + 2x3 − 2A−4x3 + A−4x + A−8x3 − A−12xS2,∞(560) = −A12x + A8x3 − 2A4x3 + A4x + 2x3 − A−4x3 + A−8xS2,∞(561) = A2x2 − A−2x4 − A−2x2 + 2A−6x4 − A−6x2 − A−10x4 − A−10x2 + 2A−14x2 − A−14 − A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − A10x4 − A10x2 + 2A6x4 − A6x2 − A2x4 − A2x2 + A−2x2

S2,∞(562) = A8x − 2A4x3 + x5 + x3 − x − A−4x5 − A−4x + 2A−8x3 − A−8x − A−12xS2,∞(562) = −A12x + 2A8x3 − A8x − A4x5 − A4x + x5 + x3 − x − 2A−4x3 + A−8xS2,∞(563) = −A6x2 + A2x4 − 2A−2x4 + A−2x2 + A−6x4 + A−6x2 − 2A−10x2 + A−10 + A−14

S2,∞(563) = A14 − 2A10x2 + A10 + A6x4 + A6x2 − 2A2x4 + A2x2 + A−2x4 − A−6x2

S2,∞(564) = A8x − 2A4x3 + A4x + x5 + x − A−4x5 + 2A−8x3 − A−8x − A−12xS2,∞(564) = −A12x + 2A8x3 − A8x − A4x5 + x5 + x − 2A−4x3 + A−4x + A−8xS2,∞(565) = −A6x2+2A2x4−A2x2−A−2x6+A−2x4−2A−2x2+3A−6x4−2A−6x2−3A−10x2+A−10+A−14


S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − A2x6 + A2x4 − 2A2x2 + 2A−2x4 − A−2x2 − A−6x2

S2,∞(566) = A10x2 − A6x4 + A6x2 + A2x4 − A−2x4 + 2A−6x2 − A−6 − A−10

S2,∞(566) = −A10 + 2A6x2 − A6 − A2x4 + A−2x4 − A−6x4 + A−6x2 + A−10x2

S2,∞(567) = −A−6x4 + 2A−6x2 + A−10x4 − A−10x2 − A−14x4 + 2A−14x2 − A−14 + A−18x2 − A−18

S2,∞(567) = A18x2 − A18 − A14x4 + 2A14x2 − A14 + A10x4 − A10x2 − A6x4 + 2A6x2

S2,∞(568) = −A8x3 + A8x + A4x5 − 2A4x3 − x5 + 3x3 − 2x + A−4x3 − 2A−4xS2,∞(568) = A4x3 − 2A4x − x5 + 3x3 − 2x + A−4x5 − 2A−4x3 − A−8x3 + A−8xS2,∞(569) = A−2x4 − 2A−2x2 − A−6x6 + 3A−6x4 − 2A−6x2 + 2A−10x4 − 4A−10x2 + A−10 − A−14x2 + A−14

S2,∞(569) = −A14x2 + A14 + 2A10x4 − 4A10x2 + A10 − A6x6 + 3A6x4 − 2A6x2 + A2x4 − 2A2x2

S2,∞(570) = A12x3 − A12x − A8x5 + 2A8x3 − A8x + A4x5 − 2A4x3 + A4x − x3 + 2xS2,∞(570) = −x3 + 2x + A−4x5 − 2A−4x3 + A−4x − A−8x5 + 2A−8x3 − A−8x + A−12x3 − A−12xS2,∞(571) = A10x4 − 3A10x2 + A10 − A6x4 + 2A6x2 − A2 + A−2x2 − A−2 − A−6

S2,∞(571) = −A6 + A2x2 − A2 − A−2 − A−6x4 + 2A−6x2 + A−10x4 − 3A−10x2 + A−10

S2,∞(572) = A−8x5 − 3A−8x3 + 2A−8x − A−12x5 + 2A−12x3 + A−16x3 − A−16xS2,∞(572) = A16x3 − A16x − A12x5 + 2A12x3 + A8x5 − 3A8x3 + 2A8xS2,∞(573) = A6x4 − 2A6x2 + A6 − A2x6 + 4A2x4 − 5A2x2 + A2 + A−2x4 − 2A−2x2

S2,∞(573) = A2x4 − 2A2x2 − A−2x6 + 4A−2x4 − 5A−2x2 + A−2 + A−6x4 − 2A−6x2 + A−6

S2,∞(574) = −A−10x4 + 3A−10x2 − A−10 + A−18x2 − A−22

S2,∞(574) = −A22 + A18x2 − A10x4 + 3A10x2 − A10

S2,∞(575) = A10x4 − 3A10x2 + A10 − A6x4 + 2A6x2 + A−2

S2,∞(575) = A2 − A−6x4 + 2A−6x2 + A−10x4 − 3A−10x2 + A−10

S2,∞(576) = A4x3 − 2A4x − x3 + x + A−4x3 − 2A−4xS2,∞(576) = A4x3 − 2A4x − x3 + x + A−4x3 − 2A−4xS2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x4 + 2A−2x2 − A−2 + A−6x2 − A−10

S2,∞(577) = −A10 + A6x2 − A2x4 + 2A2x2 − A2 + 2A−2x2 − A−2 − A−6x2 + A−6

S2,∞(578) = A4x3 − 2A4x − x3 + 2x + A−8xS2,∞(578) = A8x − x3 + 2x + A−4x3 − 2A−4xS2,∞(579) = −A−6x2 + A−6 + A−10x2 − A−14x4 + A−14x2 + A−18x4 − A−18x2 + A−18 − A−22x2

S2,∞(579) = −A22x2 + A18x4 − A18x2 + A18 − A14x4 + A14x2 + A10x2 − A6x2 + A6

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x4 + 2A2x2 − A2 + A−2x2

S2,∞(580) = A2x2 − A−2x4 + 2A−2x2 − A−2 + A−6x2 − A−10x2 + A−14x2 − A−14

S2,∞(581) = −A−10x4 + 3A−10x2 − A−10 − A−14 + 2A−18x2 − A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A10x4 + 3A10x2 − A10

S2,∞(582) = −A−6x4 + 2A−6x2 − A−6 + A−10x4 − A−10 − A−14x4 + A−14x2 + A−18x2

S2,∞(582) = A18x2 − A14x4 + A14x2 + A10x4 − A10 − A6x4 + 2A6x2 − A6

S2,∞(583) = −A4x3 + A4x + x5 − 2x3 + 2x − A−4x5 + 2A−4x3 − A−4x + A−8x3 − A−8xS2,∞(583) = A8x3 − A8x − A4x5 + 2A4x3 − A4x + x5 − 2x3 + 2x − A−4x3 + A−4xS2,∞(584) = A−2x4 − 2A−2x2 + A−2 − A−6x6 + 3A−6x4 − 3A−6x2 + A−6 + 2A−10x4 − 3A−10x2 − A−14x2

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − A6x6 + 3A6x4 − 3A6x2 + A6 + A2x4 − 2A2x2 + A2

S2,∞(585) = −A−10x6 + 5A−10x4 − 6A−10x2 + A−10 + A−14x4 − 3A−14x2 + A−14

S2,∞(585) = A14x4 − 3A14x2 + A14 − A10x6 + 5A10x4 − 6A10x2 + A10

S2,∞(586) = A14 − A10x2 + 2A6x2 − A6 − 2A2x2 + A−2x2 − A−2 − A−6

S2,∞(586) = −A6 + A2x2 − A2 − 2A−2x2 + 2A−6x2 − A−6 − A−10x2 + A−14

S2,∞(587) = −A−6x2 + A−6 + A−10x2 − A−10 − A−14x2 + A−18 + A−26

S2,∞(587) = A26 + A18 − A14x2 + A10x2 − A10 − A6x2 + A6


S2,∞(588) = −A−6x2 + A−6 + A−10x2 − 2A−14x2 + A−14 + A−18x2 − A−22 + A−26

S2,∞(588) = A26 − A22 + A18x2 − 2A14x2 + A14 + A10x2 − A6x2 + A6

S2,∞(589) = −A−6x2 + 2A−10x2 − 2A−10 − 2A−14x2 + A−18x2 − A−22 + A−26

S2,∞(589) = A26 − A22 + A18x2 − 2A14x2 + 2A10x2 − 2A10 − A6x2

S2,∞(590) = −A−2 + A−6 − A−10x2 + A−14x2 − A−14 − A−18x2 + A−22x2 − A−22

S2,∞(590) = A22x2 − A22 − A18x2 + A14x2 − A14 − A10x2 + A6 − A2

S2,∞(591) = −A−2 − A−10 − A−18x2 + A−18 + A−22x2 − A−22

S2,∞(591) = A22x2 − A22 − A18x2 + A18 − A10 − A2

S2,∞(592) = A−4x − 2A−8x + A−12x3 + A−12x − A−16x3 + 2A−20x − A−24xS2,∞(592) = −A24x + 2A20x − A16x3 + A12x3 + A12x − 2A8x + A4xS2,∞(593) = −A−10x2 + A−10 + A−30

S2,∞(593) = A30 − A10x2 + A10

S2,∞(594) = −A−6 − A−14x2 + A−18x2 − A−22x2 + A−26x2 − A−26

S2,∞(594) = A26x2 − A26 − A22x2 + A18x2 − A14x2 − A6

S2,∞(595) = −A−6 − A−14x2 − A−14 + 2A−18x2 − A−18 − 2A−22x2 + A−22 + A−26x2

S2,∞(595) = A26x2 − 2A22x2 + A22 + 2A18x2 − A18 − A14x2 − A14 − A6

Knots in L(2, 1)

S2,∞(11) = A6 − 1 − A−2 − A−4

S2,∞(21) = −A−2 − A−4 − A−6 + A−12

S2,∞(21) = −A10 + A4 − A−2 − A−4

S2,∞(22) = −A4x + x + A−8xS2,∞(23) = A−12xS2,∞(23) = −A8x + x + A−4xS2,∞(31) = A−4x + A−12x − A−16xS2,∞(31) = A12x − A8x + A−4xS2,∞(32) = A6 + A4 − A−2 − A−10 − A−12 − A−14

S2,∞(32) = 1 − A−14 − A−16 − A−18

S2,∞(33) = −A−2 − A−4 + A−10 − A−16

S2,∞(33) = A14 − A8 − A−2 − A−4

S2,∞(34) = −A4x + 2x + A−8x − A−12xS2,∞(34) = A8x − A4x + x − A−4x + A−8xS2,∞(35) = A2 + 1 − A−4 − A−6 + A−12 − A−14 − A−16 − A−18

S2,∞(35) = −A10 + A6 + 2A4 + A2 − A−2 − A−4 − A−10 − A−12 − A−14

S2,∞(36) = A8 + A6 + A4 − A−2 − A−4 − A−6 − A−8 − A−10

S2,∞(38) = −A−2 − A−10 − A−12 − A−14 + A−18 + A−20

S2,∞(41) = −A12 + A6 − A−8 − A−10

S2,∞(42) = −A8x + A−12x + A−16xS2,∞(42) = −A−8x + A−20x + A−24xS2,∞(43) = 2A−4x − A−8x + A−12x − 2A−16x + A−20xS2,∞(43) = −A16x + 2A12x − A8x + A4x − x + A−4xS2,∞(44) = A2 + 1 − 2A−4 − A−6 − A−8 − A−14 − A−16 + A−20 + A−22


S2,∞(44) = −A10 − A8 + A6 + A4 + A2 − 1 − A−2 − A−4 + A−6 + A−8 − A−12 − A−14

S2,∞(45) = −2A4x + 2x − A−4x + A−8x − A−12x + A−16x + A−20xS2,∞(45) = A8x − 2A4x + x − 2A−4x + A−8x + A−16x + A−20xS2,∞(46) = −A−6 − A−8 − A−10 + A−24

S2,∞(46) = −A22 + A18 + A16 − A10 − A2 − 1S2,∞(47) = A−4x − A−8x + 2A−12x − A−16x + A−20x − A−24xS2,∞(47) = A20x − 2A16x + A12x − A8x + 2A4xS2,∞(48) = −A16x + A12x − A8x + A4x + A−4xS2,∞(48) = −x + A−4x + 2A−12x − A−16xS2,∞(49) = A14 + A12 − A8 − A6 − A4 − A−4

S2,∞(49) = A2 − A−2 − A−4 − A−12 − A−14 − A−16 + A−18 + A−20

S2,∞(410) = −A−2 − A−4 − A−14 + A−20

S2,∞(410) = −A18 + A12 − A−2 − A−4

S2,∞(411) = −A4x + 2x − A−4x + A−8x − A−12x + A−16xS2,∞(411) = −A12x + A8x − A4x + 2x − A−4x + A−8xS2,∞(412) = A2 + 1 − A−2 − 2A−4 − A−6 + A−10 − A−14 − 2A−16 + A−20 + A−22

S2,∞(412) = A14 − A10 − 2A8 + A4 + A2 − A−2 − A−4 + A−6 + A−8 − A−12 − A−14

S2,∞(413) = A8x − 2A4x + 2x − A−4x + 2A−8x − A−12xS2,∞(414) = A8x − 3A4x + 2x − 2A−4x + 2A−8x − A−12x + A−16x + A−20xS2,∞(415) = −A6 − A4 + 1 − A−4 − A−6 + A−10 + 2A−12 − A−16 − A−18

S2,∞(415) = −A10 + A6 + 2A4 − 1 − 2A−2 − A−4 − A−10 − A−12 + A−16 + A−18

S2,∞(416) = −A16x + A12x − A8x + A4x + A−4xS2,∞(417) = A6 + A4 − 1 − A−2 − A−4 − A−8 − A−10 − A−12 + A−16 + A−18

S2,∞(417) = −A4 − A−4 + A−10 + A−12 − A−16 − A−18

S2,∞(418) = −x + A−4x − A−8x + A−12x − A−16x + A−20x + A−24xS2,∞(418) = A12x − 2A8x − x + A−4x + A−12x + A−16xS2,∞(419) = −A10 − A8 + A4 − A−2 + A−6 + A−8 − A−12 − A−14

S2,∞(419) = 1 − A−2 − A−4 − A−6 − A−14 − A−16 + A−20 + A−22

S2,∞(420) = −A−6 − A−8 − A−10 + A−12 − A−18 − A−20 + A−24 + A−26

S2,∞(420) = −A14 − A12 − A10 + A8 + A6 + A4 − A−8 − A−10

S2,∞(421) = −A4x + x − A−4x + A−16x + A−20xS2,∞(423) = A−8x + A−12x − A−20xS2,∞(424) = −A8x − A4x + A−4x + A−8x + A−12xS2,∞(426) = −A10 + A6 − 1 + A−6 − A−10 − A−12

S2,∞(51) = A−8x + A−12x − A−20xS2,∞(51) = xS2,∞(52) = A−6 − A−10 − 2A−12 − A−14 + A−18 − A−22 − A−24 + A−28 + A−30

S2,∞(53) = x − A−4x + A−8x + A−16x − A−20xS2,∞(54) = A12 + A10 + A8 − A4 − A2 − A−6 − A−8 − A−10

S2,∞(55) = A−12xS2,∞(56) = A8 + A6 + A−6 − A−16 − A−18 − A−20 − A−22 − A−24

S2,∞(56) = −A−8 + A−12 + A−14 + A−16 + A−18 − A−28 − A−30 − A−32 − A−34 − A−36

S2,∞(57) = −A−6 − 2A−8 − A−10 + A−12 − A−18 + A−22 + 2A−24 + A−26 − A−28 − A−30

S2,∞(57) = A16 − A14 − A12 − A10 + A8 + A6 − A2 − 1 + A−4 + A−6 − A−8 − A−10

S2,∞(58) = −x + A−4x − A−8x + 2A−12x + A−20x − A−28x


S2,∞(58) = A12x − 2A8x + A4x + A−4x − A−8x + A−16xS2,∞(59) = A6 + 2A4 − 1 − A−2 + A−6 − A−12 + A−16 + A−18 − A−20 − A−22 − A−24 − A−26 − A−28

S2,∞(59) = −A4 + 1−A−4 +A−8 + 2A−10 + 2A−12 +A−14 −A−16 −A−18 −A−24 −A−26 −A−28 −A−30 −A−32

S2,∞(510) = 2A−4x − 2A−8x + 2A−12x − 2A−16x + 2A−20x − A−24xS2,∞(510) = A20x − 2A16x + 2A12x − 2A8x + 2A4x − x + A−4xS2,∞(511) = A2 + 1 − A−2 − 3A−4 − A−6 + A−10 − A−14 − A−16 + A−18 + 2A−20 + A−22 − A−24 − A−26

S2,∞(511) = A14 + A12 − A10 − 2A8 + A4 + A2 − 1 − 2A−2 − 2A−4 + A−6 + 2A−8 + A−10 − A−12 − A−14

S2,∞(512) = −2x + 3A−4x − 2A−8x + 3A−12x − 2A−16x + 2A−20x − A−28xS2,∞(512) = −A16x + 3A12x − 3A8x + 2A4x − 2x + 2A−4x − A−8x + A−16xS2,∞(513) = −A6 − A4 + 2 − A−4 − A−6 + A−10 + A−12 − A−14 − 2A−16 − A−18 + A−20 + A−22

S2,∞(513) = −A10 − A8 + A6 + 2A4 − 1 − 2A−2 + A−6 + A−8 − A−10 − 2A−12 − A−14 + A−16 + A−18

S2,∞(514) = −2A4x + 3x − 2A−4x + 2A−8x − A−12x + 2A−16x − A−24xS2,∞(514) = −A12x + 2A8x − 2A4x + 3x − 2A−4x + A−8x − A−12x + A−20xS2,∞(515) = −A10 − A8 + 2A6 + 3A4 + A2 − 2− 2A−2 + 2A−6 + 2A−8 − 2A−12 − A−14 + A−16 + A−18 − A−20 −A−22 − A−24 − A−26 − A−28

S2,∞(515) = −A6 −A4 +A2 + 3+A−2 − 2A−4 −A−6 + 2A−10 + 2A−12 − 2A−16 −A−18 +A−20 +A−22 −A−24 −A−26 − A−28 − A−30 − A−32

S2,∞(516) = 2A8x − 3A4x + 3x − 2A−4x + 2A−8x − 2A−12x + A−20xS2,∞(516) = A8x − 3A4x + 3x − 2A−4x + 3A−8x − A−12x + A−16x − A−24xS2,∞(517) = A14 + A12 − A10 − A8 + 2A4 + A2 − 2A−2 − 2A−4 + A−8 − A−12 − A−14

S2,∞(517) = A2 + 1 − A−2 − 2A−4 − A−6 + A−8 + A−10 + A−12 − A−14 − A−16 + A−20 − A−24 − A−26

S2,∞(518) = −A−6 − 2A−8 + A−12 + A−14 − A−16 − A−18 − A−20 + A−22 + 2A−24 + A−26 − A−28 − A−30

S2,∞(518) = A18 + A16 − 2A12 − A10 + A6 − A2 − 1 + A−4 + A−6 − A−8 − A−10

S2,∞(519) = −2A4x + 2x − A−4x + 2A−8x + A−16x − A−24xS2,∞(519) = A8x − 2A4x + 2x − A−4x + A−8x − A−12x + A−20xS2,∞(520) = A2 + 2 − 2A−4 − A−6 + A−10 + A−12 − A−16 + A−20 + A−22 − A−24 − A−26 − A−28 − A−30 − A−32

S2,∞(520) = −A10 − A8 + A6 + 2A4 + A2 − 1 − A−2 + 2A−6 + 2A−8 + A−10 − A−12 − A−14 − A−20 − A−22 −A−24 − A−26 − A−28

S2,∞(521) = 2A12x − 2A8x + A4x − x + A−4x − A−8x + A−16xS2,∞(521) = −x + 2A−4x − A−8x + 2A−12x − A−16x + A−20x − A−28xS2,∞(522) = −A6 − A4 + 1 + A−2 − A−4 − A−8 + A−10 + A−12 − A−16 − A−18

S2,∞(522) = 2A6 + A4 − 2 − 2A−2 − A−4 − A−10 − A−12 + A−16 + A−18

S2,∞(523) = 2x − A−4x − A−12x + A−16xS2,∞(524) = −A18 + A12 + A10 + A8 + A6 − A−2 − A−4 − A−6 − A−8 − A−10

S2,∞(525) = −A−6 − A−8 + A−10 + A−12 − 2A−16 − A−18 + A−22 + A−24 − A−26 − 2A−28 + A−32 + A−34

S2,∞(525) = A26 − 2A22 − 2A20 + A18 + 2A16 + 2A14 − A10 − A8 + A6 + A4 − A2 − 2 − A−2

S2,∞(526) = −x + 3A−4x − 2A−8x + 3A−12x − 3A−16x + 2A−20x − A−24xS2,∞(526) = A20x − 3A16x + 3A12x − 2A8x + 3A4x − 2x + A−4xS2,∞(527) = −x + 3A−4x − 2A−8x + 3A−12x − 3A−16x + 2A−20x − A−24xS2,∞(527) = A20x − 3A16x + 3A12x − 2A8x + 3A4x − 2x + A−4xS2,∞(528) = A14 + A12 − A10 − A8 + A4 + A2 − A−2 − A−4 + A−8 − A−10 − A−12 − A−14

S2,∞(528) = −A6 + 2A2 + 2 − A−4 − A−6 − 2A−14 − 2A−16 + A−20

S2,∞(529) = −A6 − A4 + 1 − A−6 + A−12 − A−16

S2,∞(529) = A14 − 2A10 − A8 + A4 − A−2 + A−6 + A−8 − A−10 − A−12

S2,∞(530) = A8x − A4x + 2x − 2A−4x + A−8x − A−12x + A−16x


S2,∞(530) = −A12x + 2A8x − A4x + 2x − 2A−4x + A−8xS2,∞(531) = A8x − 2A4x + 3x − 2A−4x + 2A−8x − 2A−12x + A−16xS2,∞(531) = −A12x + 3A8x − 2A4x + 2x − 3A−4x + 2A−8xS2,∞(532) = A8x − 2A4x + 2x − 2A−4x + 2A−8x − A−12x + A−16xS2,∞(532) = −A12x + 2A8x − 2A4x + 2x − 2A−4x + 2A−8xS2,∞(533) = −A6 − A4 + 1 + A−2 − A−8 − A−16

S2,∞(533) = A14 − A10 − A8 + A6 − 1 − A−2 + A−6 + A−8 − A−10 − A−12

S2,∞(534) = −A6 − A4 + A2 + 2 + A−2 − A−6 + A−12 − A−14 − 2A−16 − A−18

S2,∞(534) = A14 − 2A10 − A8 + A6 + 2A4 + A2 − A−2 + A−6 + A−8 − 2A−10 − 2A−12 − A−14

S2,∞(535) = −A−6 − A−8 + A−12 − A−14 − A−16 − A−18 + A−20 + A−22 + A−24 − A−26 − 2A−28 + A−32 + A−34

S2,∞(535) = A26 − 2A22 − 2A20 + 2A16 + A14 + A12 − A10 + A6 + A4 − A2 − 2 − A−2

S2,∞(536) = −x + 2A−4x − 2A−8x + 3A−12x − 2A−16x + 2A−20x − A−24xS2,∞(536) = A20x − 3A16x + 2A12x − 2A8x + 3A4x − x + A−4xS2,∞(537) = A14 + A12 − A10 − A8 − A6 − A−2 − A−4 + A−8

S2,∞(537) = −A6 + A2 + 1 − A−2 − A−4 − A−6 − A−14 − A−16 + A−18 + A−20

S2,∞(538) = −A−2 + A−8 − A−10 − A−12 − A−14 + A−18 − A−22 − A−24 + A−28 + A−30

S2,∞(539) = −A4x + x − A−4x + 2A−8x + A−16x − A−20xS2,∞(540) = A−8x + A−16x − A−20x + A−24x − A−28xS2,∞(540) = A24x − A20x − A12x + A8x + xS2,∞(541) = A18 + A8 − A2 − A−6 − A−8 − A−10

S2,∞(542) = A20x − A16x + A12x − 2A8x + A4x + A−4xS2,∞(543) = A6 + A4 − 1 − A−2 − A−10 + A−16 − A−20 − A−22

S2,∞(543) = A8 + 1 − A−4 − A−6 − A−8 + A−12 − A−16 − A−18

S2,∞(544) = −x + 2A−4x − 2A−8x + 2A−12x − A−16x + 2A−20x − A−28xS2,∞(544) = −A16x + 2A12x − 2A8x + 2A4x − x + A−4x − A−8x + A−16xS2,∞(545) = −A10 − A8 + A6 + 2A4 + A2 − 1 − A−2 − A−4 + A−6 − A−10 − 2A−12 − A−14 + A−16 + A−18

S2,∞(545) = −A4 + A2 + 1 − 2A−4 − A−6 + A−10 + A−12 − A−14 − 2A−16 − A−18 + A−20 + A−22

S2,∞(546) = −x + 3A−4x − 2A−8x + 2A−12x − 2A−16x + 2A−20x − A−28xS2,∞(546) = −A16x + 3A12x − 2A8x + 2A4x − 2x + A−4x − A−8x + A−16xS2,∞(547) = −A10 − A8 + A6 + 3A4 + A2 − 1 − A−2 + 2A−6 + A−8 − 2A−12 − A−14 + A−16 + A−18 − A−20 −A−22 − A−24 − A−26 − A−28

S2,∞(547) = −A4 + A2 + 2 − 2A−4 − A−6 + A−8 + 2A−10 + 2A−12 − 2A−16 − A−18 + A−20 + A−22 − A−24 −A−26 − A−28 − A−30 − A−32

S2,∞(548) = A4x − x + 2A−4x − A−8x + A−12x − 2A−16x + A−24xS2,∞(548) = A12x − 2A8x + A4x − x + 2A−4x + A−12x − A−20xS2,∞(549) = A14 + A12 − A8 + A4 − 2A−2 − A−4 + A−8 − A−12 − A−14

S2,∞(549) = 1 − A−2 − A−4 + A−8 + A−10 − A−14 − A−16 + A−20 − A−24 − A−26

S2,∞(550) = −A−6 − A−8 − A−16 + A−22 + A−24 − A−28

S2,∞(550) = A26 − A22 − A20 + A18 + A16 + A14 − A10 − A8 − A2 − 1S2,∞(551) = A−4x − A−8x + 3A−12x − 2A−16x + A−20x − 2A−24x + A−28xS2,∞(552) = −A16x + A12x − A8x + 2A4x + A−4x − A−8xS2,∞(552) = A4x − x + A−4x − A−8x + 2A−12x − A−16xS2,∞(553) = A−2 + A−4 − A−8 − A−10 − A−18 − A−20 − A−22 + A−24

S2,∞(553) = −A22 + A18 + A16 + A8 + A6 − A2 − 1 − A−6 − A−8 − A−10

S2,∞(554) = −x + 2A−4x − A−8x + 3A−12x − 2A−16x + A−20x − A−24x


S2,∞(555) = −A18 − A16 + A14 + 2A12 + A10 − A8 − A6 − A4 + A2 + 1 − 2A−4 − A−6

S2,∞(555) = A2 −A−2 −A−4 +A−6 +A−8 − 2A−12 − 2A−14 −A−16 +A−18 +A−20 −A−22 −A−24 +A−28 +A−30

S2,∞(556) = A14 + A12 − A8 − A−2 + A−8 − A−10 − A−12 − A−14

S2,∞(556) = −A6 + A2 + 2 − A−12 − 2A−14 − 2A−16 + A−20

S2,∞(557) = −A−2 − A−4 + A−18 − A−24

S2,∞(557) = A22 − A16 − A−2 − A−4

S2,∞(558) = −A4x + 2x − A−4x + 2A−8x − A−12x + A−16x − A−20xS2,∞(558) = A16x − A12x + A8x − 2A4x + 2x − A−4x + A−8xS2,∞(559) = A2 + 1 − A−2 − 2A−4 + A−8 + A−10 − 2A−14 − A−16 + 2A−20 − A−24 − A−26

S2,∞(559) = −A18 + A14 + 2A12 − A8 + A4 + A2 − 2A−2 − 2A−4 + A−8 − A−12 − A−14

S2,∞(560) = A8x − 2A4x + 3x − 2A−4x + 2A−8x − 2A−12x + A−16xS2,∞(560) = −A12x + 2A8x − 2A4x + 3x − 2A−4x + 2A−8x − A−12xS2,∞(561) = A2 + 1− 2A−2 − 3A−4 −A−6 +A−8 +A−10 − 2A−14 −A−16 +A−18 + 3A−20 +A−22 −A−24 −A−26

S2,∞(561) = −A18 +A14 + 2A12 −A10 − 2A8 −A6 +A4 +A2 − 2A−2 − 2A−4 +A−6 + 2A−8 +A−10 −A−12 −A−14

S2,∞(562) = A8x − 4A4x + 4x − 3A−4x + 4A−8x − 2A−12x + 2A−16x − A−24xS2,∞(562) = −A12x + 3A8x − 4A4x + 4x − 3A−4x + 3A−8x − 2A−12x + A−20xS2,∞(563) = −A6 − A4 + A2 + 2 − 2A−4 − A−6 + 2A−10 + A−12 − A−14 − 3A−16 − A−18 + A−20 + A−22

S2,∞(563) = A14−A10−2A8+A6+2A4+A2−1−2A−2−A−4+A−6+A−8−A−10−2A−12−A−14+A−16+A−18

S2,∞(564) = A8x − 3A4x + 4x − 3A−4x + 3A−8x − 2A−12x + 2A−16x − A−24xS2,∞(564) = −A12x + 3A8x − 3A4x + 4x − 3A−4x + 2A−8x − 2A−12x + A−20xS2,∞(565) = −A6 − A4 + 2A2 + 4 − 3A−4 − 2A−6 + A−8 + 3A−10 + 2A−12 − A−14 − 4A−16 − A−18 + 2A−20 +2A−22 − A−24 − A−26 − A−28 − A−30 − A−32

S2,∞(565) = A14 − 2A10 − 3A8 +A6 + 4A4 + 2A2 − 1− 2A−2 + 3A−6 + 3A−8 − 3A−12 − 2A−14 +A−16 +A−18 −A−20 − A−22 − A−24 − A−26 − A−28

S2,∞(566) = A10 + A8 − A4 + 1 − A−4 − 2A−6 − A−8 + 2A−12 − A−16 − A−18

S2,∞(566) = −A10 + A6 + 2A4 − 1 − A−2 + A−6 − A−10 − A−12 + A−16 − A−20 − A−22

S2,∞(567) = A16 + A8 + A6 − A2 − 1 − A−2 − A−8 − A−10

S2,∞(568) = −A8x + A−4x + A−8x + A−12x − A−20xS2,∞(568) = A−4x − A−16x + A−24xS2,∞(569) = A−4 − A−8 + A−12 + A−14 − A−20 + A−24 + A−26 − A−28 − A−30 − A−32 − A−34 − A−36

S2,∞(569) = −A12 + A8 + A6 + 1 + A−2 + A−4 + A−6 − A−8 − A−10 − A−16 − A−18 − A−20 − A−22 − A−24

S2,∞(570) = A12x − A8x + A4x − A−8x + A−16xS2,∞(570) = A−4x − A−8x + A−12x + A−20x − A−28xS2,∞(571) = −A6 + 1 + A−4 − A−8 − A−10 − A−12 − A−14 − A−16 + A−24 + A−26

S2,∞(572) = A−8x + A−24x − A−32xS2,∞(572) = A16x − A4x + A−12xS2,∞(573) = A6 + A4 − 1 − A−2 + A−6 + A−8 + A−10 − A−20 − A−22 − A−24 − A−26 − A−28

S2,∞(573) = 1 − A−4 + A−10 + A−12 + A−14 − A−24 − A−26 − A−28 − A−30 − A−32

S2,∞(575) = A2 − A−8 − A−10 − A−12 − A−14 − A−16 + A−24 + A−26

S2,∞(576) = A−12xS2,∞(577) = −A10 + A6 + A4 + A−4 + A−6 − 2A−10 − 2A−12 − A−14

S2,∞(579) = −A−8 − A−16 − A−26 − A−28 + A−32 + A−34

S2,∞(579) = −A22 − A20 + A18 + A16 + A14 + A6 − A2 − 2 − A−2

S2,∞(580) = A12 − A−10 − A−12 − A−14

S2,∞(580) = A2 + 1 − A−10 − A−12 − 2A−14 − A−16 + A−20


S2,∞(581) = A26 − A22 − A20 + A16 + A14 + 2A12 − A2 − 1 − A−2 − A−4 − A−6

S2,∞(582) = −A−6 − A−8 + 2A−12 + A−14 − A−18 − A−20 + A−24 − A−28 − A−30

S2,∞(582) = A18 + A16 − A12 − A10 + A8 + A6 + A4 − A2 − 1 − A−2 − A−8 − A−10

S2,∞(583) = −A4x + 2x − A−4x + A−8x + A−16x − A−24xS2,∞(583) = A8x − A4x + 2x − A−4x − A−12x + A−20xS2,∞(584) = A−2+A−4−2A−8−A−10+A−12+A−14+A−16−A−20+A−24+A−26−A−28−A−30−A−32−A−34−A−36

S2,∞(584) = −A14 − A12 + 2A8 + 2A6 + A−2 + A−4 + A−6 − A−8 − A−10 − A−16 − A−18 − A−20 − A−22 − A−24

S2,∞(585) = A6 + A4 + A2 + 1 + A−2 − A−8 − A−10 − A−12 − A−14 − A−16 − A−18 − A−20

S2,∞(586) = A14 − A10 − A8 + A4 − 2A−2 − A−4 + A−8

S2,∞(586) = −A6 + 1 − A−2 − A−4 − A−6 + A−10 + A−12 − A−16

S2,∞(588) = A26 − A22 + A18 + A16 − A12 − A10 − A8 + A6 − A2 − 1S2,∞(589) = A26 − A22 + A18 + A16 − A14 − A12 − 2A10 + A6 + A4 − A2 − 1S2,∞(590) = −A−2 + A−6 − A−10 − A−12 − A−14 − A−22 + A−26 + A−28

S2,∞(592) = A−4x − 2A−8x + 3A−12x − A−16x + 2A−20x − 2A−24xS2,∞(594) = −A−6 − A−14 − A−16 − A−26 + A−30 + A−32

S2,∞(595) = −A−6 − 2A−14 − A−16 + A−20 + A−22 − A−26 − A−28 + A−30 + A−32

Knots in L(3, 1)

S2,∞(11) = −A−3xS2,∞(11) = A6 − Ax − A−2

S2,∞(21) = −A−2 − A−3x − A−6 + A−7xS2,∞(21) = −A10 + A5x − Ax − A−2

S2,∞(22) = −A4x + x + A−3 + A−7

S2,∞(23) = −A8x + A+ x + A−3

S2,∞(31) = A−4x + A−7 − A−15

S2,∞(31) = A12x − A8x − A5 + x + A−3

S2,∞(32) = A6 + A5x − Ax − A−2 − A−7xS2,∞(32) = Ax − A−3x − A−11xS2,∞(33) = −A−2 − A−3x + A−7x + A−10 − A−11xS2,∞(33) = A14 − A9x + A5x − Ax − A−2

S2,∞(34) = −A4x + 2x + A−3 − A−11

S2,∞(34) = A8x − A4x − A+ x + A−7

S2,∞(35) = A2 + Ax − 2A−3x − A−6 + A−7x − A−11xS2,∞(35) = −A10 + A6 + 2A5x + A2 − 2Ax − A−2 − A−7xS2,∞(36) = A9x + A6 − Ax − A−2 − A−3xS2,∞(37) = A18 − A5x − A2

S2,∞(38) = −A−2 − A−10 − A−11x − A−14 + A−15x + A−18

S2,∞(41) = −A−7x − A−15x + A−19xS2,∞(41) = −A13x + A9x + A6 − A−2 − A−3xS2,∞(42) = −A8x − A5 + x + A−8x + A−11

S2,∞(43) = 2A−4x + A−7 − A−8x − A−11 − A−15 + A−19

S2,∞(43) = −A16x + 2A12x + A9 − A8x − A5 − A+ x + A−3


S2,∞(44) = A2 + Ax − 3A−3x − A−6 + A−7x − A−11x + A−15xS2,∞(44) = −A10 − A9x + A6 + 2A5x + A2 − 2Ax − A−2 + A−3x − A−7xS2,∞(45) = −2A4x − A+ 2x + A−3 − A−11 + A−12x + A−15

S2,∞(45) = A8x − 2A4x − 2A+ x + A−7 + A−12x + A−15

S2,∞(46) = −A−6 − A−7x − A−10 + A−11x − A−15x + A−19xS2,∞(46) = −A22 + A18 + A17x − A13x − A10 + A9x − A5x − A2

S2,∞(47) = A−4x − A−8x + 2A−12x + A−15 − A−16x − A−23

S2,∞(47) = A20x − 2A16x − A13 + A12x + A5 + A4xS2,∞(48) = −A16x + A12x − A8x + A4x + A+ A−3

S2,∞(48) = −x + A−4x + A−7 + A−11 + A−12x − A−16xS2,∞(49) = A2 − A−2 − A−3x + A−7x − 2A−11x − A−14 + A−15x + A−18

S2,∞(410) = −A−2 − A−3x + A−7x − A−11x − A−14 + A−15xS2,∞(410) = −A18 + A13x − A9x + A5x − Ax − A−2

S2,∞(411) = −A4x + 2x + A−3 − A−4x + A−15

S2,∞(411) = −A12x + A8x + A5 − A4x + x + A−7

S2,∞(412) = A2 + Ax − A−2 − 3A−3x − A−6 + 2A−7x + A−10 − 2A−11x + A−15xS2,∞(412) = A14 − A10 − 2A9x + 3A5x + A2 − 2Ax − A−2 + A−3x − A−7xS2,∞(413) = A8x − 2A4x − A+ 2x + A−3 + A−7 − A−11

S2,∞(414) = A8x − 3A4x − 2A+ 2x + A−3 + A−7 − A−11 + A−12x + A−15

S2,∞(415) = −A6 − A5x + 2Ax − 2A−3x − A−6 + 2A−7x − A−11xS2,∞(415) = −A10 + A6 + 2A5x − 3Ax − 2A−2 + A−3x − A−7x + A−11xS2,∞(416) = A−4x + A−7 − A−8x + A−19

S2,∞(416) = −A16x + A12x + A9 − A8x + x + A−3

S2,∞(417) = A6 + A5x − 2Ax − A−2 − A−7x + A−11xS2,∞(417) = −A5x + Ax − A−3x + A−7x − A−11xS2,∞(418) = −x − A−3 + A−4x + A−7 − A−15 + A−16x + A−19

S2,∞(418) = A12x − 2A8x − 2A5 + x + A−3 + A−8x + A−11

S2,∞(419) = −A10 − A9x + 2A5x − Ax − A−2 + A−3x − A−7xS2,∞(419) = Ax − A−2 − 2A−3x − A−6 + A−7x − A−11x + A−15xS2,∞(420) = −A14 − A13x − A10 + 2A9x + A6 − A−2 − A−3xS2,∞(421) = −A4x − A+ x + A−12x + A−15

S2,∞(422) = −A16x + A5 + A4x + AS2,∞(423) = A−8x + A−11 + A−12x − A−16x − A−19

S2,∞(424) = −A9 − A8x − A5 + x + A−3 + A−4x + A−7

S2,∞(425) = −A10 − A2 + A−3x + A−6 − A−7x − A−10

S2,∞(426) = −A10 + A6 − Ax + A−3x + A−6 − A−7x − A−10

S2,∞(427) = A8x − 2A4x + x + A−3 − A−4x + A−7 + A−8xS2,∞(51) = A−8x + A−11 + A−12x − A−16x − A−19

S2,∞(51) = −A9 + A4x + AS2,∞(52) = A−6 − A−10 − 2A−11x − A−14 + A−15x + A−18 − A−19x + A−23xS2,∞(53) = −A12x − A9 + A8x + A5 + A4x + A− A−3

S2,∞(53) = x − A−4x − A−7 + A−8x + A−11 + A−12x + A−15 − A−16x − A−19

S2,∞(54) = A13x + A10 − A5x − A2 − A−3xS2,∞(55) = A13 − A8x + A−4x + A−7

S2,∞(56) = A9x + A6 − A−2 − A−10 − A−11x − A−14


S2,∞(57) = −A−6 − 2A−7x − A−10 + 2A−11x − A−15x + 2A−19x − A−23xS2,∞(57) = A17x − A14 − 2A13x − A10 + 2A9x + A6 − A5x + Ax − A−2 − A−3xS2,∞(58) = −x − A−3 + A−4x + 2A−7 + A−11 + A−16x + A−19 − A−20x − A−23

S2,∞(58) = A12x + A9 − 2A8x − A5 + A+ x + A−3 − A−4x − A−7 + A−8x + A−11

S2,∞(59) = A6 + 2A5x − 2Ax − A−2 + A−3x − A−7x + A−11x − A−14 − A−15x − A−18

S2,∞(59) = −A5x + 2Ax − A−3x + 2A−7x − A−11x − A−18 − A−19x − A−22

S2,∞(510) = 2A−4x + A−7 − 2A−8x − A−11 + A−12x + A−19 − A−23

S2,∞(510) = A20x − 2A16x − A13 + 2A12x + A9 − A8x − A+ x + A−3

S2,∞(511) = A2 + Ax − A−2 − 4A−3x − A−6 + 3A−7x + A−10 − 2A−11x + 2A−15x − A−19xS2,∞(511) = A14 + A13x − A10 − 3A9x + 3A5x + A2 − 3Ax − A−2 + 2A−3x − A−7xS2,∞(512) = −2x − A−3 + 3A−4x + 3A−7 − A−8x − A−15 + A−16x + 2A−19 − A−20x − A−23

S2,∞(512) = −A16x + 3A12x + 2A9 − 3A8x − 2A5 + x + 2A−3 − A−4x − A−7 + A−8x + A−11

S2,∞(513) = −A6 − A5x + 3Ax − 3A−3x − A−6 + 2A−7x − 2A−11x + A−15xS2,∞(513) = −A10 − A9x + A6 + 3A5x − 3Ax − 2A−2 + 2A−3x − 2A−7x + A−11xS2,∞(514) = −2A4x − A+ 3x + 2A−3 − A−4x + A−12x + 2A−15 − A−16x − A−19

S2,∞(514) = −A12x + 2A8x + 2A5 − 2A4x − A+ x + A−7 − A−8x − A−11 + A−12x + A−15

S2,∞(515) = −A10 −A9x + 2A6 + 4A5x +A2 − 4Ax − 2A−2 + 3A−3x − 2A−7x +A−11x −A−14 −A−15x −A−18

S2,∞(515) = −A6 −A5x +A2 + 4Ax +A−2 − 4A−3x −A−6 + 3A−7x − 2A−11x +A−15x −A−18 −A−19x −A−22

S2,∞(516) = 2A8x + A5 − 3A4x − 2A+ 2x + A−3 + A−7 − A−8x − 2A−11 + A−12x + A−15

S2,∞(516) = A8x − 3A4x − 2A+ 3x + 2A−3 + A−7 − A−11 + A−12x + A−15 − A−16x − A−19

S2,∞(517) = A14 + A13x − A10 − 2A9x + 3A5x + A2 − 3Ax − A−2 + A−3x − A−7xS2,∞(517) = A2 + Ax − A−2 − 3A−3x − A−6 + 3A−7x + A−10 − 2A−11x + A−15x − A−19xS2,∞(518) = −A−6 − 2A−7x + 2A−11x + A−14 − 2A−15x + 2A−19x − A−23xS2,∞(518) = A18 + A17x − 3A13x − A10 + 2A9x + A6 − A5x + Ax − A−2 − A−3xS2,∞(519) = −2A4x − A+ 2x + 2A−3 + A−7 + A−12x + A−15 − A−16x − A−19

S2,∞(519) = A8x + A5 − 2A4x − A+ x + A−3 + A−7 − A−8x − A−11 + A−12x + A−15

S2,∞(520) = A2 + 2Ax − 3A−3x − A−6 + 2A−7x − A−11x + A−15x − A−18 − A−19x − A−22

S2,∞(520) = −A10 − A9x + A6 + 3A5x + A2 − 2Ax − A−2 + 2A−3x − A−7x − A−14 − A−15x − A−18

S2,∞(521) = 2A12x + A9 − 2A8x − 2A5 + x + A−3 − A−4x − A−7 + A−8x + A−11

S2,∞(521) = −x − A−3 + 2A−4x + 2A−7 − A−15 + A−16x + A−19 − A−20x − A−23

S2,∞(522) = −A6 − A5x + 2Ax + A−2 − 2A−3x + A−7x − A−11xS2,∞(522) = 2A6 + A5x − 3Ax − 2A−2 + A−3x − A−7x + A−11xS2,∞(523) = −A12x + A8x + A5 + x − A−3

S2,∞(523) = 2x − A−4x − A−7 + A−15

S2,∞(524) = −A18 + A13x + A10 + A6 − Ax − A−2 − A−3xS2,∞(525) = −A−6 − A−7x + A−10 + 2A−11x − 4A−15x − A−18 + 3A−19x + A−22 − 2A−23x + A−27xS2,∞(525) = A26 − 2A22 − 2A21x + A18 + 4A17x + 2A14 − 3A13x − A10 + 2A9x − 2A5x − A2

S2,∞(526) = −x + 3A−4x + A−7 − 2A−8x − A−11 + 2A−12x − A−16x + A−19 − A−23

S2,∞(526) = A20x − 3A16x − A13 + 3A12x + A9 − A8x + A4x − A+ A−3

S2,∞(527) = −x + 3A−4x + A−7 − 2A−8x − A−11 + 2A−12x − A−16x + A−19 − A−23

S2,∞(527) = A20x − 3A16x − A13 + 3A12x + A9 − A8x + A4x − A+ A−3

S2,∞(528) = A14 + A13x − A10 − 2A9x + 3A5x + A2 − 3Ax − A−2 + A−3x − A−7xS2,∞(528) = −A6 + 2A2 + 2Ax − 3A−3x − A−6 + 2A−7x − 3A−11x − A−14 + A−15x + A−18

S2,∞(529) = A14 − 2A10 − A9x + 2A5x − 2Ax − A−2 + 2A−3x + A−6 − A−7x − A−10

S2,∞(530) = A8x − A4x + 2x − 2A−4x − A−7 + A−8x + A−15


S2,∞(530) = −A12x + 2A8x + A5 − A4x + x − A−3 − A−4x + A−8xS2,∞(531) = A8x − 2A4x + 3x + A−3 − 2A−4x − A−7 + A−8x − A−11 + A−15

S2,∞(531) = −A12x + 3A8x + A5 − 2A4x − A+ x − A−3 − A−4x + A−7 + A−8xS2,∞(532) = A8x − 2A4x + 2x + A−3 − 2A−4x + A−8x + A−15

S2,∞(532) = −A12x + 2A8x + A5 − 2A4x + x − A−4x + A−7 + A−8xS2,∞(533) = A14 − A10 − A9x + A6 + A5x − 2Ax − A−2 + 2A−3x + A−6 − A−7x − A−10

S2,∞(534) = −A6 − A5x + A2 + 3Ax + A−2 − 3A−3x − A−6 + 2A−7x − 2A−11xS2,∞(534) = A14 − 2A10 − A9x + A6 + 3A5x + A2 − 3Ax − A−2 + 2A−3x + A−6 − 2A−7x − A−10

S2,∞(535) = −A−6 − A−7x + 2A−11x − A−14 − 3A−15x − A−18 + 3A−19x + A−22 − 2A−23x + A−27xS2,∞(535) = A26 − 2A22 − 2A21x + 4A17x + A14 − 2A13x − A10 + 2A9x − 2A5x − A2

S2,∞(536) = −x + 2A−4x + A−7 − 2A−8x + 2A−12x + A−15 − A−16x + A−19 − A−23

S2,∞(536) = A20x − 3A16x − A13 + 2A12x + A9 − A8x + A5 + A4x + A−3

S2,∞(537) = A14 + A13x − A10 − 2A9x − A6 + 2A5x − 2Ax − A−2 + A−3xS2,∞(537) = −A6 + A2 + Ax − A−2 − 2A−3x − A−6 + 2A−7x − 2A−11x − A−14 + A−15x + A−18

S2,∞(538) = −A−2 + A−7x − A−10 − 2A−11x − A−14 + A−15x + A−18 − A−19x + A−23xS2,∞(539) = −A12x − A9 + A8x + A5 + A+ A−7

S2,∞(539) = −A4x + x + A−3 − A−4x + A−8x + A−11 + A−12x + A−15 − A−16x − A−19

S2,∞(540) = A−8x + A−11 − A−27

S2,∞(540) = A24x − A20x − A17 + A4x + AS2,∞(541) = A18 + A9x − A5x − A2 − A−3xS2,∞(542) = A−4x + A−7 − A−8x + A−12x − A−23

S2,∞(542) = A20x − A16x − A13 + A12x − A8x + x + A−3

S2,∞(543) = A6 + A5x − 2Ax − A−2 + A−3x − A−7x + A−11x − A−15xS2,∞(543) = A9x − A5x + Ax − 2A−3x + A−7x − A−11xS2,∞(544) = −x − A−3 + 2A−4x + 2A−7 − A−8x + A−16x + 2A−19 − A−20x − A−23

S2,∞(544) = −A16x + 2A12x + 2A9 − 2A8x − A5 + x + A−3 − A−4x − A−7 + A−8x + A−11

S2,∞(545) = −A10 − A9x + A6 + 3A5x + A2 − 3Ax − A−2 + A−3x − 2A−7x + A−11xS2,∞(545) = −A5x + A2 + 2Ax − 3A−3x − A−6 + 2A−7x − 2A−11x + A−15xS2,∞(546) = −x − A−3 + 3A−4x + 2A−7 − A−8x − A−11 − A−15 + A−16x + 2A−19 − A−20x − A−23

S2,∞(546) = −A16x + 3A12x + 2A9 − 2A8x − 2A5 − A+ x + A−3 − A−4x − A−7 + A−8x + A−11

S2,∞(547) = −A10 − A9x + A6 + 4A5x + A2 − 3Ax − A−2 + 2A−3x − 2A−7x + A−11x − A−14 − A−15x − A−18

S2,∞(547) = −A5x + A2 + 3Ax − 3A−3x − A−6 + 3A−7x − 2A−11x + A−15x − A−18 − A−19x − A−22

S2,∞(548) = A4x + A− x − A−3 + A−4x + A−7 − A−12x − 2A−15 + A−16x + A−19

S2,∞(548) = A12x − 2A8x − 2A5 + A4x + A+ x + A−3 + A−8x + A−11 − A−12x − A−15

S2,∞(549) = A14 + A13x − 2A9x + 2A5x − 2Ax − A−2 + A−3x − A−7xS2,∞(549) = Ax − A−2 − 2A−3x + 2A−7x + A−10 − 2A−11x + A−15x − A−19xS2,∞(550) = −A−6 − A−7x + A−11x − 2A−15x + 2A−19x + A−22 − A−23xS2,∞(550) = A26 − A22 − A21x + A18 + 2A17x + A14 − 2A13x − A10 + A9x − A5x − A2

S2,∞(551) = A−4x − A−8x + 3A−12x + A−15 − 2A−16x − A−19 − A−23 + A−27

S2,∞(551) = −A24x + 2A20x + A17 − 2A16x − A13 + A12x − A9 + A5 + A4xS2,∞(552) = −A16x + A12x − A8x + 2A4x + A− A−7

S2,∞(552) = A4x − x − A−3 + A−4x + A−11 + A−12x − A−16xS2,∞(553) = A−2 + A−3x − 2A−7x − A−10 + A−11x − 2A−15x + A−19xS2,∞(553) = −A22 + A18 + A17x − A13x + 2A9x + A6 − 2A5x − A2 − A−3xS2,∞(554) = A20x − 2A16x − A13 + 2A12x − A8x + A4x + A−3


S2,∞(554) = −x + 2A−4x + A−7 − A−8x + 2A−12x − A−16x − A−23

S2,∞(555) = A2 − A−2 − A−3x + A−6 + 2A−7x − 4A−11x − 2A−14 + 2A−15x + A−18 − A−19x + A−23xS2,∞(556) = A14 + A13x − 2A9x + 2A5x − 2Ax − A−2 + A−3x − A−7xS2,∞(556) = −A6 + A2 + 2Ax − 2A−3x + A−7x − 3A−11x − A−14 + A−15x + A−18

S2,∞(557) = −A−2 − A−3x + A−7x − A−11x + A−15x + A−18 − A−19xS2,∞(557) = A22 − A17x + A13x − A9x + A5x − Ax − A−2

S2,∞(558) = −A4x + 2x + A−3 − A−4x + A−8x − A−19

S2,∞(558) = A16x − A12x − A9 + A8x − A4x + x + A−7

S2,∞(559) = A2 + Ax − A−2 − 3A−3x + 3A−7x + A−10 − 3A−11x − A−14 + 2A−15x − A−19xS2,∞(559) = −A18 + A14 + 2A13x − 3A9x + 3A5x + A2 − 3Ax − A−2 + A−3x − A−7xS2,∞(560) = A8x − 2A4x − A+ 3x + A−3 − A−4x − A−11 + A−15

S2,∞(560) = −A12x + 2A8x + A5 − 2A4x − A+ 2x + A−7 − A−11

S2,∞(561) = A2 + Ax − 2A−2 − 4A−3x − A−6 + 4A−7x + A−10 − 3A−11x − A−14 + 3A−15x − A−19xS2,∞(561) = −A18 + A14 + 2A13x − A10 − 4A9x − A6 + 4A5x + A2 − 3Ax − A−2 + 2A−3x − A−7xS2,∞(562) = A8x − 4A4x − 2A+ 4x + 3A−3 − A−4x + A−7 − A−11 + A−12x + 2A−15 − A−16x − A−19

S2,∞(562) = −A12x + 3A8x + 2A5 − 4A4x − 2A+ 2x + A−3 + 2A−7 − A−8x − 2A−11 + A−12x + A−15

S2,∞(563) = −A6 − A5x + A2 + 3Ax − 4A−3x − A−6 + 3A−7x + A−10 − 3A−11x + A−15xS2,∞(563) = A14 − A10 − 2A9x + A6 + 4A5x + A2 − 4Ax − 2A−2 + 2A−3x − 2A−7x + A−11xS2,∞(564) = A8x − 3A4x − 2A+ 4x + 2A−3 − A−4x − A−11 + A−12x + 2A−15 − A−16x − A−19

S2,∞(564) = −A12x + 3A8x + 2A5 − 3A4x − 2A+ 2x + A−7 − A−8x − 2A−11 + A−12x + A−15

S2,∞(565) = −A6−A5x+2A2+5Ax−6A−3x−2A−6+5A−7x+A−10−4A−11x+2A−15x−A−18−A−19x−A−22

S2,∞(565) = A14−2A10−3A9x+A6+7A5x+2A2−5Ax−2A−2+4A−3x−3A−7x+A−11x−A−14−A−15x−A−18

S2,∞(566) = A10 + A9x − 2A5x + 2Ax − 3A−3x − A−6 + 2A−7x − A−11xS2,∞(566) = −A10 + A6 + 2A5x − 3Ax − A−2 + 2A−3x + A−6 − 2A−7x + A−11x − A−15xS2,∞(567) = A17x − A13x + A9x + A6 − A5x − A−2 − A−3xS2,∞(568) = −A8x − A5 + A+ x + A−3 + A−7 + A−8x + A−11 − A−12x − A−15

S2,∞(569) = A−3x − A−7x + A−11x − A−15x + A−19x − A−22 − A−23x − A−26

S2,∞(569) = −A13x + 2A9x + A6 + Ax − A−2 − A−3x − A−10 − A−11x − A−14

S2,∞(570) = A12x + A9 − A8x − A5 + x − A−4x − A−7 + A−8x + A−11

S2,∞(570) = −A−3 + A−4x + A−7 + A−16x + A−19 − A−20x − A−23

S2,∞(571) = −A6 + Ax − A−7x − A−10 − A−11x − A−14 + A−18 + A−19xS2,∞(572) = A−8x + A−11 − A−15 + A−20x + A−23 − A−24x − A−27

S2,∞(572) = A16x + A13 − A9 − A8x − A5 + A−4x + A−7

S2,∞(573) = A6 + A5x − Ax − A−2 + A−3x − A−14 − A−15x − A−18

S2,∞(573) = Ax − A−3x + A−7x − A−18 − A−19x − A−22

S2,∞(574) = −A22 + A18 + A17x + A14 − A5x − A2 − AxS2,∞(575) = A2 − A−7x − A−10 − A−11x − A−14 + A−18 + A−19xS2,∞(576) = A− A−4x + A−8x + A−11

S2,∞(577) = −A10 + A6 + A5x − Ax + A−3x + A−6 − 2A−7x − A−10

S2,∞(578) = A8x − A−3 − A−4x + A−8x + A−11

S2,∞(579) = −A−7x + A−11x − 2A−15x + A−19x − A−23x + A−27xS2,∞(579) = −A22 − A21x + A18 + 2A17x + A14 − A13x + A9x − 2A5x − A2

S2,∞(580) = A13x − A9x + A5x − Ax − A−7xS2,∞(580) = A2 + Ax − A−3x − A−10 − 2A−11x − A−14 + A−15x + A−18

S2,∞(581) = A26 − A22 − A21x + 2A17x + A14 − A5x − A2 − Ax


S2,∞(582) = A18 + A17x − 2A13x − A10 + 2A9x + A6 − A5x − A−2 − A−3xS2,∞(583) = −A4x − A+ 2x + A−3 + A−12x + A−15 − A−16x − A−19

S2,∞(583) = A8x + A5 − A4x − A+ x − A−8x − A−11 + A−12x + A−15

S2,∞(584) = A−2 + A−3x − 2A−7x − A−10 + 2A−11x − A−15x + A−19x − A−22 − A−23x − A−26

S2,∞(584) = −A14 − A13x + 3A9x + 2A6 − A5x + Ax − A−2 − A−3x − A−10 − A−11x − A−14

S2,∞(585) = A9x + A6 + A5x − A−2 − A−3x − A−6 − A−7x − A−10

S2,∞(586) = A14 − A10 − A9x + 2A5x − 2Ax − 2A−2 + A−3xS2,∞(586) = −A6 + Ax − A−2 − 2A−3x − A−6 + 2A−7x + A−10 − A−11xS2,∞(587) = A26 + A18 − A14 − A13x − A10 + A9x + A6 − A5x − A2

S2,∞(588) = A26 − A22 + A18 + A17x − 2A13x − A10 + A9x + A6 − A5x − A2

S2,∞(589) = −A−6 − A−7x − A−10 + 2A−11x − 2A−15x − A−18 + A−19x + A−26

S2,∞(589) = A26 − A22 + A18 + A17x − A14 − 2A13x − 2A10 + 2A9x + A6 − A5x − A2

S2,∞(590) = −A−2 + A−6 − A−10 − A−11x − A−14 + A−15x − A−19x − A−22 + A−23x + A−26

S2,∞(591) = −A−2 − A−10 − A−19x − A−22 + A−23x + A−26

S2,∞(592) = A−4x − 2A−8x + 3A−12x + A−15 − A−16x + A−20x − A−23 − A−24xS2,∞(592) = −A24x + 2A20x − 2A16x − A13 + 2A12x − A8x + A5 + A4xS2,∞(593) = A30 − A9x − A6

S2,∞(594) = −A−6 − A−14 − A−15x + A−19x − A−23x − A−26 + A−27x + A−30

S2,∞(595) = −A−6 − 2A−14 − A−15x + 2A−19x + A−22 − 2A−23x − A−26 + A−27x + A−30

S2,∞(595) = A26 + A25x − 2A21x − A18 + 2A17x − A13x − A10 − A6

Knots in L(4, 1)

S2,∞(22) = −A4x + x + A−2xS2,∞(23) = A−6xS2,∞(23) = −A8x + A2x + xS2,∞(31) = A−4x + A−6x − A−10xS2,∞(31) = A12x − A8x − A6x + A2x + xS2,∞(32) = A6x2 − A2x2 − A−2 − A−6

S2,∞(32) = A2x2 − A2 − A−2x2 − A−10

S2,∞(34) = −A4x + 2x + A−2x − A−6xS2,∞(34) = A8x − A4x − A2x + x + A−2xS2,∞(35) = A2x2 − 2A−2x2 + A−2 + A−6x2 − A−6 − 2A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 − A−6

S2,∞(36) = A10x2 − A10 − A2x2 − A−2

S2,∞(41) = −A14x2 + A14 + A10x2 + A6 − A2x2 + A2 − A−2

S2,∞(42) = −A8x − A6x + A2x + x + A−6xS2,∞(42) = −A−2x + A−6x + A−14xS2,∞(43) = 2A−4x + A−6x − A−8x − 2A−10x + A−14xS2,∞(43) = −A16x + 2A12x + A10x − A8x − 2A6x + A2x + xS2,∞(44) = A2x2 − 3A−2x2 + 2A−2 + A−6x2 − A−10 + A−14

S2,∞(44) = −A10x2 + 2A6x2 − 2A2x2 + 2A2 + A−2 − A−6

S2,∞(45) = −2A4x − A2x + 2x + 2A−2x − A−6x + A−10x


S2,∞(45) = A8x − 2A4x − 2A2x + x + 2A−2x + A−10xS2,∞(47) = A−4x − A−8x + 2A−12x + A−14x − A−16x − A−18xS2,∞(47) = A20x − 2A16x − A14x + A12x + A10x + A4xS2,∞(48) = −A16x + A12x − A8x + A4x + A2xS2,∞(48) = −x + A−4x + A−6x + A−12x − A−16xS2,∞(411) = −A4x + 2x + A−2x − A−4x − A−6x + A−10xS2,∞(411) = −A12x + A8x + A6x − A4x − A2x + x + A−2xS2,∞(412) = A2x2 − 3A−2x2 + A−2 + 2A−6x2 − A−6 − A−10x2 + 2A−14

S2,∞(412) = A14 − 2A10x2 + A10 + 3A6x2 − A6 − 2A2x2 + A2 + A−2 − A−6

S2,∞(413) = A8x − 2A4x − A2x + 2x + 2A−2x − A−6xS2,∞(414) = A8x − 3A4x − 2A2x + 2x + 3A−2x − A−6x + A−10xS2,∞(415) = −A6x2 + 2A2x2 − A2 − 2A−2x2 + A−2 + A−6x2 − 2A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 + A−2x2 − A−2 − A−6 + A−10

S2,∞(416) = A−4x + A−6x − A−8x − A−10x + A−14xS2,∞(416) = −A16x + A12x + A10x − A8x − A6x + A2x + xS2,∞(417) = A6x2 − 2A2x2 + A2 + A−10

S2,∞(417) = −A6x2 + A6 + A2x2 − A−2x2 + A−2 + A−6 − A−10

S2,∞(418) = −x − A−2x + A−4x + 2A−6x − A−10x + A−14xS2,∞(418) = A12x − 2A8x − 2A6x + 2A2x + x + A−6xS2,∞(419) = −A10x2 + 2A6x2 − A6 − A2x2 − A−6

S2,∞(419) = A2x2 − A2 − 2A−2x2 + A−6x2 − A−6 − A−10 + A−14

S2,∞(420) = −A−6x2 + A−10x2 − 2A−10 − A−14 + A−18

S2,∞(420) = −A14x2 + 2A10x2 − 2A10 − A2x2 + A2 − A−2

S2,∞(421) = −A4x − A2x + x + A−2x + A−10xS2,∞(422) = −A16x + A6x + A4xS2,∞(423) = A−8x + A−10x + A−12x − A−14x − A−16xS2,∞(424) = −A10x − A8x + A2x + x + A−2xS2,∞(427) = A8x − 2A4x + x + A−2x − A−4x + A−8xS2,∞(51) = A−8x + A−10x + A−12x − A−14x − A−16xS2,∞(51) = −A10x + A6x + A4xS2,∞(52) = A−6 − 2A−10x2 + A−10 + A−14x2 − A−14 + A−22

S2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 − A10x2 + A10 − A6x2 + 2A6 − A2

S2,∞(53) = −A12x − A10x + A8x + 2A6x + A4x − A2xS2,∞(53) = x − A−4x − A−6x + A−8x + 2A−10x + A−12x − A−14x − A−16xS2,∞(54) = A14x2 − A14 − A6x2 + A6 − A2 − A−2

S2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 − A−10x2 + A−14x2 − 2A−14

S2,∞(55) = A−10x + A−18x − A−22xS2,∞(55) = A14x − A10x − A8x + x + A−2xS2,∞(56) = A10x2 − A10 + A6 − A2x2 + A2 − A−6x2 + A−6

S2,∞(57) = −2A−6x2 + A−6 + 2A−10x2 − 2A−10 − A−14 + A−18 − A−22

S2,∞(57) = A18x2 − A18 − 2A14x2 + 2A10x2 − 2A10 − A2x2 + 2A2 − A−2

S2,∞(58) = −x − A−2x + A−4x + 3A−6x − A−10x + A−14x − A−18xS2,∞(58) = A12x + A10x − 2A8x − 2A6x + 2A2x + x − A−2x + A−6xS2,∞(59) = 2A6x2 − A6 − 2A2x2 + A2 − A−10x2 + 2A−10

S2,∞(59) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 + A−2 + A−6 − A−10 − A−14x2 + A−14


S2,∞(510) = 2A−4x + A−6x − 2A−8x − 2A−10x + A−12x + 2A−14x − A−18xS2,∞(510) = A20x − 2A16x − A14x + 2A12x + 2A10x − A8x − 2A6x + A2x + xS2,∞(511) = A2x2 − 4A−2x2 + 2A−2 + 3A−6x2 − A−6 − A−10x2 + 2A−14 − A−18

S2,∞(511) = A14x2 − 3A10x2 + A10 + 3A6x2 − A6 − 2A2x2 + A2 + 2A−2 − A−6

S2,∞(512) = −2x − A−2x + 3A−4x + 4A−6x − A−8x − 3A−10x + 2A−14x − A−18xS2,∞(512) = −A16x + 3A12x + 2A10x − 3A8x − 4A6x + 3A2x + x − A−2x + A−6xS2,∞(513) = −A6x2 + 3A2x2 − 2A2 − 3A−2x2 + A−2 + A−6x2 − 2A−10 + A−14

S2,∞(513) = −A10x2 + 3A6x2 − A6 − 3A2x2 + A2 + A−2x2 − A−2 − 2A−6 + A−10

S2,∞(514) = −2A4x − A2x + 3x + 3A−2x − A−4x − 2A−6x + 2A−10x − A−14xS2,∞(514) = −A12x + 2A8x + 2A6x − 2A4x − 3A2x + x + 2A−2x − A−6x + A−10xS2,∞(515) = −A10x2 + 4A6x2 − A6 − 4A2x2 + 3A2 + A−2x2 − 2A−6 − A−10x2 + 2A−10

S2,∞(515) = −A6x2 + 4A2x2 − 2A2 − 4A−2x2 + 3A−2 + A−6x2 + A−6 − 2A−10 − A−14x2 + 2A−14

S2,∞(516) = 2A8x + A6x − 3A4x − 3A2x + 2x + 3A−2x − 2A−6x + A−10xS2,∞(516) = A8x − 3A4x − 2A2x + 3x + 4A−2x − 2A−6x + A−10x − A−14xS2,∞(517) = A14x2 − 2A10x2 + 3A6x2 − 2A6 − 2A2x2 + A−2 − A−6

S2,∞(517) = A2x2 − 3A−2x2 + A−2 + 3A−6x2 − 2A−6 − A−10x2 − A−10 + A−14 − A−18

S2,∞(518) = −2A−6x2 + A−6 + 2A−10x2 − A−10 − A−14x2 + A−14 + 2A−18 − A−22

S2,∞(518) = A18x2 − 3A14x2 + 2A14 + 2A10x2 − A10 − A2x2 + 2A2 − A−2

S2,∞(519) = −2A4x − A2x + 2x + 3A−2x − A−6x + A−10x − A−14xS2,∞(519) = A8x + A6x − 2A4x − 2A2x + x + 2A−2x − A−6x + A−10xS2,∞(520) = 2A2x2 − A2 − 3A−2x2 + 2A−2 + A−6x2 − A−10 − A−14x2 + 2A−14

S2,∞(520) = −A10x2 + 3A6x2 − A6 − 2A2x2 + 2A2 + A−2 − A−6 − A−10x2 + A−10

S2,∞(521) = 2A12x + A10x − 2A8x − 3A6x + 2A2x + x − A−2x + A−6xS2,∞(521) = −x − A−2x + 2A−4x + 3A−6x − 2A−10x + A−14x − A−18xS2,∞(522) = −A6x2 + 2A2x2 − A2 − 2A−2x2 + 2A−2 + 2A−6 − A−10

S2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 + A−2x2 − A−2 − A−6 + A−10

S2,∞(523) = −A12x + A8x + A6x − A2x + xS2,∞(523) = 2x − A−4x − A−6x + A−10xS2,∞(524) = A−6 − A−10x2 + A−14x2 − 2A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 + A6 − A2x2 − A−2

S2,∞(525) = −A−6x2 + 2A−10x2 − 4A−14x2 + 2A−14 + 3A−18x2 − A−18 − A−22x2 − A−22 + 2A−26

S2,∞(525) = A26 − 2A22x2 + 4A18x2 − A18 − 3A14x2 + 2A14 + A10x2 + A10 − A6x2 − A6

S2,∞(526) = −x + 3A−4x + A−6x − 2A−8x − 2A−10x + 2A−12x + 2A−14x − A−16x − A−18xS2,∞(526) = A20x − 3A16x − A14x + 3A12x + 2A10x − A8x − 2A6x + A4x + A2xS2,∞(527) = −x + 3A−4x + A−6x − 2A−8x − 2A−10x + 2A−12x + 2A−14x − A−16x − A−18xS2,∞(527) = A20x − 3A16x − A14x + 3A12x + 2A10x − A8x − 2A6x + A4x + A2xS2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 + A−2x2 − 2A−6

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 + 2A−6x2 − A−6 − 2A−10x2 − A−10 + A−14x2

S2,∞(530) = A8x − A4x + 2x − 2A−4x − A−6x + A−8x + A−10xS2,∞(530) = −A12x + 2A8x + A6x − A4x − A2x + x − A−4x + A−8xS2,∞(531) = A8x − 2A4x + 3x + A−2x − 2A−4x − 2A−6x + A−8x + A−10xS2,∞(531) = −A12x + 3A8x + A6x − 2A4x − 2A2x + x + A−2x − A−4x + A−8xS2,∞(532) = A8x − 2A4x + 2x + A−2x − 2A−4x − A−6x + A−8x + A−10xS2,∞(532) = −A12x + 2A8x + A6x − 2A4x − A2x + x + A−2x − A−4x + A−8xS2,∞(534) = −A6x2 + 3A2x2 − A2 − 3A−2x2 + A−2 + 2A−6x2 − A−6 − A−10x2 − 2A−10 + A−14


S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 + 2A−2x2 − A−2 − A−6x2 − A−6

S2,∞(535) = −A−6x2 + 2A−10x2 − A−10 − 3A−14x2 + 3A−18x2 − 2A−18 − A−22x2 − A−22 + 2A−26

S2,∞(535) = A26 − 2A22x2 + 4A18x2 − 2A18 − 2A14x2 + A10x2 − A6x2 − A6

S2,∞(536) = −x + 2A−4x + A−6x − 2A−8x − A−10x + 2A−12x + 2A−14x − A−16x − A−18xS2,∞(536) = A20x − 3A16x − A14x + 2A12x + 2A10x − A8x − A6x + A4x + A2xS2,∞(538) = −A−2 + A−6x2 − A−6 − 2A−10x2 + A−14x2 − A−14 + A−22

S2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 − A10x2 − 2A2

S2,∞(539) = −A4x + x + A−2x − A−4x − A−6x + A−8x + 2A−10x + A−12x − A−14x − A−16xS2,∞(540) = A−8x + A−10x − A−14x + A−18x − A−22xS2,∞(540) = A24x − A20x − A18x + A14x − A10x + A6x + A4xS2,∞(541) = A18 + A10x2 − A10 − A6x2 − A2 − A−2

S2,∞(541) = A−2x2 − A−2 − A−6x2 − A−10 − A−14 + A−18

S2,∞(542) = A−4x + A−6x − A−8x − A−10x + A−12x + A−14x − A−18xS2,∞(542) = A20x − A16x − A14x + A12x + A10x − A8x − A6x + A2x + xS2,∞(543) = A6x2 − 2A2x2 + A2 + A−2x2 − A−2 − A−6 − A−14

S2,∞(543) = A10x2 − A10 − A6x2 + A2x2 − A2 − A−2x2 + A−6 − A−10

S2,∞(544) = −x − A−2x + 2A−4x + 3A−6x − A−8x − 2A−10x + 2A−14x − A−18xS2,∞(544) = −A16x + 2A12x + 2A10x − 2A8x − 3A6x + 2A2x + x − A−2x + A−6xS2,∞(545) = −A10x2 + 3A6x2 − A6 − 3A2x2 + 2A2 + A−2 − A−6 + A−10

S2,∞(545) = −A6x2 + A6 + 2A2x2 − 3A−2x2 + 2A−2 + A−6x2 − 2A−10 + A−14

S2,∞(546) = −x − A−2x + 3A−4x + 3A−6x − A−8x − 3A−10x + 2A−14x − A−18xS2,∞(546) = −A16x + 3A12x + 2A10x − 2A8x − 4A6x + 2A2x + x − A−2x + A−6xS2,∞(547) = −A10x2 + 4A6x2 − 2A6 − 3A2x2 + 2A2 + A−2 − A−6 − A−10x2 + 2A−10

S2,∞(547) = −A6x2 + A6 + 3A2x2 − A2 − 3A−2x2 + 2A−2 + A−6x2 − 2A−10 − A−14x2 + 2A−14

S2,∞(548) = A4x + A2x − x − 2A−2x + A−4x + 2A−6x − 2A−10x + A−14xS2,∞(548) = A12x − 2A8x − 2A6x + A4x + 3A2x + x − A−2x + A−6x − A−10xS2,∞(549) = A14x2 − 2A10x2 + A10 + 2A6x2 − A6 − A2x2 − A2 − A−6

S2,∞(549) = A2x2 − A2 − 2A−2x2 + 2A−6x2 − A−6 − A−10x2 + A−14 − A−18

S2,∞(551) = A−4x − A−8x + 3A−12x + A−14x − 2A−16x − 2A−18x + A−22xS2,∞(551) = −A24x + 2A20x + A18x − 2A16x − 2A14x + A12x + A10x + A4xS2,∞(552) = −A16x + A12x − A8x + 2A4x + A2x − A−2xS2,∞(552) = A4x − x − A−2x + A−4x + A−6x + A−12x − A−16xS2,∞(553) = A−2x2 − 2A−6x2 + A−6 + A−10x2 − A−10 − A−14x2 − A−14 + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 − A−2

S2,∞(554) = A20x − 2A16x − A14x + 2A12x + A10x − A8x − A6x + A4x + A2xS2,∞(554) = −x + 2A−4x + A−6x − A−8x − A−10x + 2A−12x + A−14x − A−16x − A−18xS2,∞(555) = −A18x2 + 3A14x2 − A14 − 3A10x2 + 2A10 + A6x2 + A6 − A2x2 − A2 + A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − 4A−10x2 + 2A−10 + 2A−14x2 − A−14 − A−18 + A−22

S2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 + A−2x2 − A−2 − 2A−6

S2,∞(556) = −A6 + 2A2x2 − A2 − 2A−2x2 + A−6x2 − 2A−10x2 + A−14x2

S2,∞(558) = −A4x + 2x + A−2x − A−4x − A−6x + A−8x + A−10x − A−14xS2,∞(558) = A16x − A12x − A10x + A8x + A6x − A4x − A2x + x + A−2xS2,∞(559) = A2x2 − 3A−2x2 + A−2 + 3A−6x2 − A−6 − 2A−10x2 + A−14x2 − 2A−18

S2,∞(559) = −A18 + 2A14x2 − A14 − 3A10x2 + A10 + 3A6x2 − A6 − 2A2x2 + A−2 − A−6

S2,∞(560) = A8x − 2A4x − A2x + 3x + 2A−2x − A−4x − 2A−6x + A−10x


S2,∞(560) = −A12x + 2A8x + A6x − 2A4x − 2A2x + 2x + 2A−2x − A−6xS2,∞(561) = A2x2 − 4A−2x2 + A−2 + 4A−6x2 − 2A−6 − 2A−10x2 + A−14x2 + A−14 − 2A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 + 4A6x2 − 2A6 − 2A2x2 + 2A−2 − A−6

S2,∞(562) = A8x − 4A4x − 2A2x + 4x + 5A−2x − A−4x − 3A−6x + 2A−10x − A−14xS2,∞(562) = −A12x + 3A8x + 2A6x − 4A4x − 4A2x + 2x + 4A−2x − 2A−6x + A−10xS2,∞(563) = −A6x2 + 3A2x2 − A2 − 4A−2x2 + 2A−2 + 2A−6x2 − A−10x2 − A−10 + 2A−14

S2,∞(563) = A14 − 2A10x2 + A10 + 4A6x2 − A6 − 4A2x2 + 2A2 + A−2x2 − 2A−6 + A−10

S2,∞(564) = A8x − 3A4x − 2A2x + 4x + 4A−2x − A−4x − 3A−6x + 2A−10x − A−14xS2,∞(564) = −A12x + 3A8x + 2A6x − 3A4x − 4A2x + 2x + 3A−2x − 2A−6x + A−10xS2,∞(565) = −A6x2 + 5A2x2 − 2A2 − 6A−2x2 + 3A−2 + 3A−6x2 − A−6 − A−10x2 − 2A−10 − A−14x2 + 4A−14

S2,∞(565) = A14 − 3A10x2 + A10 + 7A6x2 − 3A6 − 5A2x2 + 3A2 + A−2x2 + A−2 − 3A−6 − A−10x2 + 2A−10

S2,∞(566) = A10x2 − 2A6x2 + A6 + 2A2x2 − A2 − 2A−2x2 + A−6x2 − 2A−10

S2,∞(566) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 + 2A−2x2 − A−2 − A−6x2 + A−10 − A−14

S2,∞(567) = A18x2 − A18 − A14x2 + A10x2 − A10 − A2x2 + A2 − A−2

S2,∞(568) = −A8x − A6x + 2A2x + x + A−6x − A−10xS2,∞(568) = A2x − A−2x + A−6x − A−10x + A−14xS2,∞(569) = A−2x2 − A−2 − A−6x2 + A−6 − A−18x2 + 2A−18

S2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 + A6 − A2x2 + A2 − A−2 − A−6x2 + A−6

S2,∞(570) = A12x + A10x − A8x − 2A6x + A2x + x − A−2x + A−6xS2,∞(570) = −A−2x + A−4x + 2A−6x − A−10x + A−14x − A−18xS2,∞(571) = −A6 + A2x2 − A2 − A−2 − A−6x2 + A−6 − A−10x2 + A−14x2 − A−14 + A−18

S2,∞(572) = A16x + A14x − 2A10x − A8x + x + A−2xS2,∞(573) = A6x2 − A2x2 + A2 − A−10x2 + A−10

S2,∞(573) = A2x2 − A2 − A−2x2 + A−2 + A−6 − A−14x2 + A−14

S2,∞(574) = −A22 + A18x2 − A6x2 − A2

S2,∞(575) = A2 − A−6x2 + A−6 − A−10x2 + A−14x2 − A−14 + A−18

S2,∞(576) = A2x − A−2x − A−4x + A−6x + A−8xS2,∞(577) = −A10 + A6x2 − A2x2 + A−2x2 − A−2 − A−6x2

S2,∞(578) = A8x − A−2x − A−4x + A−6x + A−8xS2,∞(579) = −A−6x2 + A−6 + A−10x2 − 2A−14x2 + A−14 + A−18x2 − A−22 + A−26

S2,∞(579) = −A22x2 + 2A18x2 − A14x2 + A14 + A10 − A6x2

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 − A−6

S2,∞(580) = A2x2 − A−2x2 − A−10x2 − A−10 + A−14x2 − A−14

S2,∞(581) = −A−14x2 − A−14 + 2A−18x2 − 2A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A6x2 − A2

S2,∞(582) = −A−6x2 + 2A−10x2 − 2A−10 − A−14x2 + A−18 − A−22

S2,∞(582) = A18x2 − 2A14x2 + A14 + 2A10x2 − 2A10 − A6 − A2x2 + A2 − A−2

S2,∞(583) = −A4x − A2x + 2x + 2A−2x − A−6x + A−10x − A−14xS2,∞(583) = A8x + A6x − A4x − 2A2x + x + A−2x − A−6x + A−10xS2,∞(584) = A−2x2 − 2A−6x2 + 2A−6 + A−10x2 − A−10 − A−14 − A−18x2 + 2A−18

S2,∞(584) = −A14x2 + 3A10x2 − 2A10 − A6x2 + 2A6 − A2x2 + 2A2 − A−2 − A−6x2 + A−6

S2,∞(585) = A10x2 + A6 − A2x2 + A2 − A−2x2

S2,∞(592) = A−4x − 2A−8x + 3A−12x + A−14x − A−16x − A−18x + A−20x − A−24xS2,∞(592) = −A24x + 2A20x − 2A16x − A14x + 2A12x + A10x − A8x + A4x


Knots in L(5, 1)

S2,∞(22) = −A4x + x + A−1x2 − A−1 − A−5

S2,∞(23) = −A8x + A3x2 − A3 + x − A−1

S2,∞(31) = A−4x + A−5x2 − A−5 − A−9x2 + A−13

S2,∞(31) = A12x − A8x − A7x2 + A7 + A3x2 + x − A−1

S2,∞(32) = A6x2 − A2x2 − A−1xS2,∞(32) = A2x2 − A2 − A−2x2 − A−5x + A−6

S2,∞(34) = −A4x + 2x + A−1x2 − A−1 − A−5x2 + A−9

S2,∞(34) = A8x − A4x − A3x2 + A3 + x + A−1x2 − A−5

S2,∞(35) = A2x2 − 2A−2x2 + A−2 − A−5x + A−6x2 − A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 − A−1x + A−2

S2,∞(36) = −A−9xS2,∞(36) = A10x2 − A10 − A3x − A2x2 + A2

S2,∞(41) = −A−6x2 + A−6 − A−9x + A−10 + A−13xS2,∞(41) = −A14x2 + A14 + A10x2 + A7x − A3x − A2x2 + A2

S2,∞(42) = −A8x − A7x2 + A7 + A3x2 + x + A−5

S2,∞(43) = 2A−4x + A−5x2 − A−5 − A−8x − 2A−9x2 + A−9 + A−13x2 + A−13 − A−17

S2,∞(43) = −A16x + 2A12x + A11x2 − A11 − A8x − 2A7x2 + A7 + A3x2 + A3 + x − A−1

S2,∞(44) = A2x2 − 3A−2x2 + 2A−2 − A−5x + A−6x2 + A−6 + A−9x − A−10

S2,∞(44) = −A10x2 + 2A6x2 + A3x − 2A2x2 + A2 − A−1x + A−2

S2,∞(45) = −2A4x − A3x2 + A3 + 2x + 2A−1x2 − A−1 − A−5x2 + 2A−9

S2,∞(45) = A8x − 2A4x − 2A3x2 + 2A3 + x + 2A−1x2 − A−5 + A−9

S2,∞(47) = A−4x − A−8x + 2A−12x + A−13x2 − A−13 − A−16x − A−17x2 + A−21

S2,∞(47) = A20x − 2A16x − A15x2 + A15 + A12x + A11x2 − A7 + A4xS2,∞(48) = −A16x + A12x − A8x + A4x + A3x2 − A3 − A−1

S2,∞(48) = −x + A−4x + A−5x2 − A−5 − A−9 + A−12x − A−16xS2,∞(411) = −A4x + 2x + A−1x2 − A−1 − A−4x − A−5x2 + A−9x2 − A−13

S2,∞(411) = −A12x + A8x + A7x2 − A7 − A4x − A3x2 + x + A−1x2 − A−5

S2,∞(412) = A2x2 − 3A−2x2 + A−2 − A−5x + 2A−6x2 + A−9x − A−10x2 + A−14

S2,∞(412) = A14 − 2A10x2 + A10 + 3A6x2 − A6 + A3x − 2A2x2 − A−1x + A−2

S2,∞(413) = A8x − 2A4x − A3x2 + A3 + 2x + 2A−1x2 − A−1 − A−5x2 − A−5 + A−9

S2,∞(414) = A8x − 3A4x − 2A3x2 + 2A3 + 2x + 3A−1x2 − A−1 − A−5x2 − A−5 + 2A−9

S2,∞(415) = −A6x2 + 2A2x2 − A2 + A−1x − 2A−2x2 − A−5x + A−6x2 − A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − A−1x + A−2x2 + A−5x − A−6

S2,∞(416) = A−4x + A−5x2 − A−5 − A−8x − A−9x2 + A−13x2 − A−17

S2,∞(416) = −A16x + A12x + A11x2 − A11 − A8x − A7x2 + A3x2 + x − A−1

S2,∞(417) = A6x2 − 2A2x2 + A2 − A−1x + A−2 + A−5xS2,∞(417) = −A6x2 + A6 + A2x2 + A−1x − A−2x2 − A−5x + A−6

S2,∞(418) = −x − A−1x2 + A−1 + A−4x + 2A−5x2 − A−5 − A−9x2 + 2A−13

S2,∞(418) = A12x − 2A8x − 2A7x2 + 2A7 + 2A3x2 + x − A−1 + A−5

S2,∞(419) = −A10x2 + 2A6x2 − A6 + A3x − A2x2 − A2 − A−1xS2,∞(419) = A2x2 − A2 − 2A−2x2 − A−5x + A−6x2 + A−9x − A−10

S2,∞(420) = −A−6x2 − A−9x + A−10x2 − A−10 + A−13x − A−14

S2,∞(420) = −A14x2 + 2A10x2 − 2A10 + A7x − A6 − A3x − A2x2 + A2


S2,∞(421) = −A4x − A3x2 + A3 + x + A−1x2 + A−9

S2,∞(422) = −A16x + A7x2 − A7 + A4x − A3

S2,∞(423) = A−8x + A−9x2 − A−9 + A−12x − A−13x2 − A−16x + A−17

S2,∞(424) = −A11x2 + A11 − A8x + A7 + A3x2 + xS2,∞(427) = A8x − 2A4x + x + A−1x2 − A−1 − A−4x − A−5 + A−8xS2,∞(51) = A−8x + A−9x2 − A−9 + A−12x − A−13x2 − A−16x + A−17

S2,∞(51) = −A11x2 + A11 + A7x2 + A4x − A3

S2,∞(52) = A−6 − 2A−10x2 + A−10 − A−13x + A−14x2 + A−17xS2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 + A11x − A10x2 − A7x − A6x2 + 2A6

S2,∞(53) = −A12x − A11x2 + A11 + A8x + 2A7x2 − A7 + A4x − A3x2 − A3 + A−1

S2,∞(53) = x − A−4x − A−5x2 + A−5 + A−8x + 2A−9x2 − A−9 + A−12x − A−13x2 − A−13 − A−16x + A−17

S2,∞(54) = A14x2 − A14 − A6x2 + A6 − A3xS2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 − A−9x − A−10x2 + A−10 + A−14x2 − A−14

S2,∞(55) = A15x2 − A15 − A11x2 − A8x + A3x2 − A3 + xS2,∞(56) = A10x2 − A10 + A7x − A3x − A2x2 + A2 − A−5xS2,∞(56) = A−5x − A−9x − A−17xS2,∞(57) = −2A−6x2 + A−6 − A−9x + 2A−10x2 − A−10 + 2A−13x − 2A−14 − A−17xS2,∞(57) = A18x2 − A18 − 2A14x2 − A11x + 2A10x2 − A10 + 2A7x − A6 − A3x − A2x2 + A2

S2,∞(58) = −x − A−1x2 + A−1 + A−4x + 3A−5x2 − 2A−5 − A−9x2 − A−9 + A−13 − A−17

S2,∞(58) = A12x + A11x2 − A11 − 2A8x − 2A7x2 + A7 + 2A3x2 − A3 + x − 2A−1 + A−5

S2,∞(59) = 2A6x2 − A6 + A3x − 2A2x2 − 2A−1x + A−2 + A−5x − A−9xS2,∞(59) = −A6x2 + A6 + 2A2x2 − A2 + 2A−1x − A−2x2 − A−2 − 2A−5x + A−6 − A−13xS2,∞(510) = 2A−4x + A−5x2 − A−5 − 2A−8x − 2A−9x2 + A−9 + A−12x + 2A−13x2 − A−17x2 − A−17 + A−21

S2,∞(510) = A20x − 2A16x − A15x2 + A15 + 2A12x + 2A11x2 − A11 − A8x − 2A7x2 + A3x2 + A3 + x − A−1

S2,∞(511) = A2x2 − 4A−2x2 + 2A−2 − A−5x + 3A−6x2 + 2A−9x − A−10x2 − A−10 − A−13x + A−14

S2,∞(511) = A14x2 − 3A10x2 + A10 − A7x + 3A6x2 + 2A3x − 2A2x2 − A−1x + A−2

S2,∞(512) = −2x − A−1x2 + A−1 + 3A−4x + 4A−5x2 − 3A−5 − A−8x − 3A−9x2 + A−13x2 + 2A−13 − 2A−17

S2,∞(512) = −A16x + 3A12x + 2A11x2 − 2A11 − 3A8x − 4A7x2 + 2A7 + 3A3x2 + x − 3A−1 + A−5

S2,∞(513) = −A6x2 + 3A2x2 − 2A2 + A−1x − 3A−2x2 − 2A−5x + A−6x2 + A−6 + A−9x − A−10

S2,∞(513) = −A10x2 + 3A6x2 − A6 + A3x − 3A2x2 − 2A−1x + A−2x2 + A−5x − A−6

S2,∞(514) = −2A4x − A3x2 + A3 + 3x + 3A−1x2 − 2A−1 − A−4x − 2A−5x2 + A−9x2 + A−9 − 2A−13

S2,∞(514) = −A12x + 2A8x + 2A7x2 − 2A7 − 2A4x − 3A3x2 + A3 + x + 2A−1x2 − 2A−5 + A−9

S2,∞(515) = −A10x2 + 4A6x2 − A6 + 2A3x − 4A2x2 + A2 − 3A−1x + A−2x2 + A−2 + A−5x − A−6 − A−9xS2,∞(515) = −A6x2+4A2x2−2A2+2A−1x −4A−2x2+A−2−3A−5x +A−6x2+2A−6+A−9x −A−10−A−13xS2,∞(516) = 2A8x + A7x2 − A7 − 3A4x − 3A3x2 + 2A3 + 2x + 3A−1x2 − A−1 − A−5x2 − 2A−5 + 2A−9

S2,∞(516) = A8x − 3A4x − 2A3x2 + 2A3 + 3x + 4A−1x2 − 2A−1 − 2A−5x2 − A−5 + 2A−9 − A−13

S2,∞(517) = A14x2 − 2A10x2 − A7x + 3A6x2 − A6 + A3x − 2A2x2 − A−1x + A−2

S2,∞(517) = A2x2 − 3A−2x2 + A−2 − A−5x + 3A−6x2 − A−6 + A−9x − A−10x2 − A−10 − A−13x + A−14

S2,∞(518) = −2A−6x2 + A−6 − A−9x + 2A−10x2 + 2A−13x − A−14x2 − A−17x + A−18

S2,∞(518) = A18x2 − 3A14x2 + 2A14 − A11x + 2A10x2 + 2A7x − A6 − A3x − A2x2 + A2

S2,∞(519) = −2A4x − A3x2 + A3 + 2x + 3A−1x2 − 2A−1 − A−5x2 − A−5 + A−9 − A−13

S2,∞(519) = A8x + A7x2 − A7 − 2A4x − 2A3x2 + A3 + x + 2A−1x2 − A−1 − 2A−5 + A−9

S2,∞(520) = 2A2x2 − A2 + A−1x − 3A−2x2 + A−2 − 2A−5x + A−6x2 + A−6 + A−9x − A−10 − A−13xS2,∞(520) = −A10x2 + 3A6x2 − A6 + 2A3x − 2A2x2 − 2A−1x + A−2 − A−9xS2,∞(521) = 2A12x + A11x2 − A11 − 2A8x − 3A7x2 + 2A7 + 2A3x2 + x − 2A−1 + A−5


S2,∞(521) = −x − A−1x2 + A−1 + 2A−4x + 3A−5x2 − 2A−5 − 2A−9x2 + 2A−13 − A−17

S2,∞(522) = −A6x2 + 2A2x2 − A2 + A−1x − 2A−2x2 + A−2 − A−5x + 2A−6

S2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 − A−1x + A−2x2 + A−5x − A−6

S2,∞(523) = −A12x + A8x + A7x2 − A7 − A3x2 + x + A−1

S2,∞(523) = 2x − A−4x − A−5x2 + A−5 + A−9x2 − A−13

S2,∞(524) = A−6 − A−9x − A−10x2 + A−10 + A−14x2 − A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 + A6 − A3x − A2x2 + A2

S2,∞(525) = −A−6x2 + 2A−10x2 − 4A−14x2 + 2A−14 − A−17x + 3A−18x2 + A−21x − A−22x2 − A−22 + A−26

S2,∞(525) = A26 − 2A22x2 + 4A18x2 − A18 + A15x − 3A14x2 + A14 − A11x + A10x2 + A10 − A6x2

S2,∞(526) = −x + 3A−4x + A−5x2 − A−5 − 2A−8x − 2A−9x2 + A−9 + 2A−12x + 2A−13x2 − A−16x − A−17x2 −A−17 + A−21

S2,∞(526) = A20x − 3A16x − A15x2 + A15 + 3A12x + 2A11x2 − A11 − A8x − 2A7x2 + A4x + A3x2 + A3 − A−1

S2,∞(527) = −x + 3A−4x + A−5x2 − A−5 − 2A−8x − 2A−9x2 + A−9 + 2A−12x + 2A−13x2 − A−16x − A−17x2 −A−17 + A−21

S2,∞(527) = A20x − 3A16x − A15x2 + A15 + 3A12x + 2A11x2 − A11 − A8x − 2A7x2 + A4x + A3x2 + A3 − A−1

S2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 − A−1x + A−2x2 + A−2 − A−6

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 − A−5x + 2A−6x2 − 2A−10x2 + A−14x2

S2,∞(530) = A8x − A4x + 2x − 2A−4x − A−5x2 + A−5 + A−8x + A−9x2 − A−13

S2,∞(530) = −A12x + 2A8x + A7x2 − A7 − A4x − A3x2 + x + A−1 − A−4x + A−8xS2,∞(531) = A8x − 2A4x + 3x + A−1x2 − A−1 − 2A−4x − 2A−5x2 + A−5 + A−8x + A−9x2 + A−9 − A−13

S2,∞(531) = −A12x + 3A8x + A7x2 − A7 − 2A4x − 2A3x2 + A3 + x + A−1x2 + A−1 − A−4x − A−5 + A−8xS2,∞(532) = A8x − 2A4x + 2x + A−1x2 − A−1 − 2A−4x − A−5x2 + A−8x + A−9x2 − A−13

S2,∞(532) = −A12x + 2A8x + A7x2 − A7 − 2A4x − A3x2 + x + A−1x2 − A−4x − A−5 + A−8xS2,∞(534) = −A6x2 + 3A2x2 − A2 − 3A−2x2 + A−2 − A−5x + 2A−6x2 − A−10x2 − A−10 + A−14

S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 − A−1x + 2A−2x2 − A−6x2

S2,∞(535) = −A−6x2 + 2A−10x2 −A−10 − 3A−14x2 −A−17x + 3A−18x2 −A−18 +A−21x −A−22x2 −A−22 +A−26

S2,∞(535) = A26 − 2A22x2 + 4A18x2 − 2A18 + A15x − 2A14x2 − A14 − A11x + A10x2 − A6x2

S2,∞(536) = −x + 2A−4x + A−5x2 − A−5 − 2A−8x − A−9x2 + 2A−12x + 2A−13x2 − A−13 − A−16x − A−17x2 −A−17 + A−21

S2,∞(536) = A20x − 3A16x − A15x2 + A15 + 2A12x + 2A11x2 − A11 − A8x − A7x2 − A7 + A4x + A3x2 − A−1

S2,∞(538) = −A−2 + A−6x2 − A−6 − 2A−10x2 − A−13x + A−14x2 + A−17xS2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 + A11x − A10x2 − A10 − A7x − A2

S2,∞(539) = −A12x − A11x2 + A11 + A8x + 2A7x2 − A7 − A3x2 − A3 + A−1x2 − A−5

S2,∞(539) = −A4x + x + A−1x2 − A−1 − A−4x − A−5x2 + A−8x + 2A−9x2 − A−9 + A−12x − A−13x2 − A−13 −A−16x + A−17

S2,∞(540) = A−8x + A−9x2 − A−9 − A−13x2 + A−17x2 − A−21x2 + A−25

S2,∞(540) = A24x − A20x − A19x2 + A19 + A15x2 − A11x2 + A7x2 + A4x − A3

S2,∞(541) = A18 + A10x2 − A10 − A6x2 − A3xS2,∞(541) = A−2x2 − A−2 − A−6x2 − A−9x + A−18

S2,∞(542) = A−4x + A−5x2 − A−5 − A−8x − A−9x2 + A−12x + A−13x2 − A−17x2 + A−21

S2,∞(542) = A20x − A16x − A15x2 + A15 + A12x + A11x2 − A8x − A7x2 + A3x2 + x − A−1

S2,∞(543) = A6x2 − 2A2x2 + A2 − A−1x + A−2x2 + A−5x − A−6 − A−9xS2,∞(543) = A10x2 − A10 − A6x2 − A3x + A2x2 + A−1x − A−2x2 − A−5x + A−6

S2,∞(544) = −x − A−1x2 + A−1 + 2A−4x + 3A−5x2 − 2A−5 − A−8x − 2A−9x2 + A−13x2 + A−13 − 2A−17

S2,∞(544) = −A16x + 2A12x + 2A11x2 − 2A11 − 2A8x − 3A7x2 + A7 + 2A3x2 + x − 2A−1 + A−5


S2,∞(545) = −A10x2 + 3A6x2 − A6 + A3x − 3A2x2 + A2 − 2A−1x + 2A−2 + A−5xS2,∞(545) = −A6x2 + A6 + 2A2x2 + A−1x − 3A−2x2 + A−2 − 2A−5x + A−6x2 + A−6 + A−9x − A−10

S2,∞(546) = −x −A−1x2 +A−1 + 3A−4x + 3A−5x2 − 2A−5 −A−8x − 3A−9x2 +A−9 +A−13x2 + 2A−13 − 2A−17

S2,∞(546) = −A16x + 3A12x + 2A11x2 − 2A11 − 2A8x − 4A7x2 + 2A7 + 2A3x2 + A3 + x − 2A−1 + A−5

S2,∞(547) = −A10x2 + 4A6x2 − 2A6 + 2A3x − 3A2x2 − 3A−1x + 2A−2 + A−5x − A−9xS2,∞(547) = −A6x2 + A6 + 3A2x2 − A2 + 2A−1x − 3A−2x2 − 3A−5x + A−6x2 + A−6 + A−9x − A−10 − A−13xS2,∞(548) = A4x + A3x2 − A3 − x − 2A−1x2 + A−1 + A−4x + 2A−5x2 − A−5 − A−9x2 − A−9 + 2A−13

S2,∞(548) = A12x − 2A8x − 2A7x2 + 2A7 + A4x + 3A3x2 − A3 + x − A−1x2 − A−1 + A−5 − A−9

S2,∞(549) = A14x2 − 2A10x2 + A10 − A7x + 2A6x2 + A3x − A2x2 − A2 − A−1xS2,∞(549) = A2x2 − A2 − 2A−2x2 − A−5x + 2A−6x2 + A−9x − A−10x2 − A−13x + A−14

S2,∞(551) = A−4x − A−8x + 3A−12x + A−13x2 − A−13 − 2A−16x − 2A−17x2 + A−17 + A−21x2 + A−21 − A−25

S2,∞(551) = −A24x + 2A20x + A19x2 − A19 − 2A16x − 2A15x2 + A15 + A12x + A11x2 + A11 − A7 + A4xS2,∞(552) = −A16x + A12x − A8x + 2A4x + A3x2 − A3 − A−1x2 + A−5

S2,∞(552) = A4x − x − A−1x2 + A−1 + A−4x + A−5x2 − A−9 + A−12x − A−16xS2,∞(553) = A−2x2 − 2A−6x2 + A−6 − A−9x + A−10x2 − A−14x2 + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 − A3x + A2

S2,∞(554) = A20x − 2A16x − A15x2 + A15 + 2A12x + A11x2 − A8x − A7x2 + A4x + A3x2 − A−1

S2,∞(554) = −x + 2A−4x + A−5x2 − A−5 − A−8x − A−9x2 + 2A−12x + A−13x2 − A−16x − A−17x2 + A−21

S2,∞(555) = −A18x2 + 3A14x2 − A14 + A11x − 3A10x2 + A10 − A7x + A6x2 + A6 − A2x2 + A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − 4A−10x2 + 2A−10 − A−13x + 2A−14x2 + A−17x − A−18

S2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 − A−1x + A−2x2 − A−6

S2,∞(556) = −A6 + 2A2x2 − A2 − 2A−2x2 − A−5x + A−6x2 + A−6 − 2A−10x2 + A−10 + A−14x2

S2,∞(558) = −A4x + 2x + A−1x2 − A−1 − A−4x − A−5x2 + A−8x + A−9x2 − A−13x2 + A−17

S2,∞(558) = A16x − A12x − A11x2 + A11 + A8x + A7x2 − A4x − A3x2 + x + A−1x2 − A−5

S2,∞(559) = A2x2 − 3A−2x2 + A−2 − A−5x + 3A−6x2 + A−9x − 2A−10x2 − A−13x + A−14x2 − A−18

S2,∞(559) = −A18 + 2A14x2 − A14 − 3A10x2 + A10 − A7x + 3A6x2 + A3x − 2A2x2 − A−1x + A−2

S2,∞(560) = A8x − 2A4x − A3x2 + A3 + 3x + 2A−1x2 − A−1 − A−4x − 2A−5x2 + A−9x2 + A−9 − A−13

S2,∞(560) = −A12x + 2A8x + A7x2 − A7 − 2A4x − 2A3x2 + A3 + 2x + 2A−1x2 − A−5x2 − A−5 + A−9

S2,∞(561) = A2x2−4A−2x2+A−2−A−5x+4A−6x2−A−6+2A−9x−2A−10x2−A−10−A−13x+A−14x2−A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 − A7x + 4A6x2 − A6 + 2A3x − 2A2x2 − A2 − A−1x + A−2

S2,∞(562) = A8x−4A4x−2A3x2+2A3+4x+5A−1x2−3A−1−A−4x−3A−5x2−A−5+A−9x2+2A−9−2A−13

S2,∞(562) = −A12x+3A8x+2A7x2−2A7−4A4x−4A3x2+2A3+2x+4A−1x2−A−1−A−5x2−3A−5+2A−9

S2,∞(563) = −A6x2 + 3A2x2 −A2 +A−1x − 4A−2x2 +A−2 − 2A−5x + 2A−6x2 +A−6 +A−9x −A−10x2 +A−14

S2,∞(563) = A14 − 2A10x2 + A10 + 4A6x2 − A6 + A3x − 4A2x2 + A2 − 2A−1x + A−2x2 + A−2 + A−5x − A−6

S2,∞(564) = A8x − 3A4x − 2A3x2 + 2A3 + 4x + 4A−1x2 − 2A−1 − A−4x − 3A−5x2 + A−9x2 + 2A−9 − 2A−13

S2,∞(564) = −A12x + 3A8x + 2A7x2 − 2A7 − 3A4x − 4A3x2 + 2A3 + 2x + 3A−1x2 − A−5x2 − 2A−5 + 2A−9

S2,∞(565) = −A6x2 + 5A2x2 − 2A2 + 2A−1x − 6A−2x2 + A−2 − 4A−5x + 3A−6x2 + A−6 + 2A−9x − A−10x2 −A−10 − A−13x + A−14

S2,∞(565) = A14−3A10x2+A10+7A6x2−3A6+3A3x−5A2x2−4A−1x+A−2x2+2A−2+A−5x−A−6−A−9xS2,∞(566) = A10x2 − 2A6x2 + A6 − A3x + 2A2x2 + A−1x − 2A−2x2 − A−5x + A−6x2 − A−10

S2,∞(566) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − A−1x + 2A−2x2 + A−5x − A−6x2 − A−9x + A−10

S2,∞(567) = −A−6x2 + A−6 − A−9x + A−10x2 + A−13x − A−14 − A−17xS2,∞(567) = A18x2 − A18 − A14x2 − A11x + A10x2 + A7x − A3x − A2x2 + A2

S2,∞(568) = −A8x − A7x2 + A7 + 2A3x2 − A3 + x − A−1 − A−9

S2,∞(569) = A−2x2 − A−2 + A−5x − A−6x2 − 2A−9x + A−10 + A−13x − A−17x


S2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 + 2A7x − A6 − 2A3x − A2x2 + A2 − A−5xS2,∞(570) = A12x + A11x2 − A11 − A8x − 2A7x2 + A7 + A3x2 + x − A−1 + A−5

S2,∞(570) = −A−1x2 + A−1 + A−4x + 2A−5x2 − A−5 − A−9x2 + A−13 − A−17

S2,∞(571) = A7x − A3x − A2x2 + A−2x2 − A−2 − A−6

S2,∞(571) = −A6 + A2x2 − A2 − A−2 − A−6x2 + A−6 − A−9x − A−10x2 + A−10 + A−13x + A−14x2 − A−14

S2,∞(572) = A−8x + A−9x2 − A−9 − A−13x2 + A−13 + A−17 − A−21

S2,∞(572) = A16x + A15x2 − A15 − 2A11x2 + A11 − A8x + A7 + A3x2 − A3 + xS2,∞(573) = A6x2 + A3x − A2x2 − A−1x − A−9xS2,∞(573) = A2x2 − A2 + A−1x − A−2x2 − A−5x + A−6 − A−13xS2,∞(574) = −A22 + A18x2 − A7x − A6x2 + A6

S2,∞(575) = A2 − A−6x2 + A−6 − A−9x − A−10x2 + A−10 + A−13x + A−14x2 − A−14

S2,∞(576) = A3x2 − A3 − A−1x2 − A−4x + A−5x2 + A−8x − A−9

S2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 − A−5x + A−6 − A−10

S2,∞(577) = −A10 + A6x2 − A2x2 − A−1x + A−2x2 − A−6x2 + A−6

S2,∞(578) = A8x − A−1x2 + A−1 − A−4x + A−5x2 + A−8x − A−9

S2,∞(579) = −A−6x2 + A−6 + A−10x2 − 2A−14x2 + A−14 − A−17x + A−18x2 + A−18 + A−21x − A−22

S2,∞(579) = −A22x2 + 2A18x2 + A15x − A14x2 − A11x + A10 − A6x2 + A6

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 − A−1x + A−2

S2,∞(580) = A2x2 − A−2x2 − A−5x + A−6 − A−10x2 + A−14x2 − A−14

S2,∞(581) = −A−13x − A−14x2 + 2A−18x2 − A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A7x − A6x2 + A6

S2,∞(582) = −A−6x2 − A−9x + 2A−10x2 − A−10 + A−13x − A−14x2 − A−17x + A−18

S2,∞(582) = A18x2 − 2A14x2 + A14 − A11x + 2A10x2 − A10 + A7x − A6 − A3x − A2x2 + A2

S2,∞(583) = −A4x − A3x2 + A3 + 2x + 2A−1x2 − A−1 − A−5x2 + A−9 − A−13

S2,∞(583) = A8x + A7x2 − A7 − A4x − 2A3x2 + A3 + x + A−1x2 − A−5 + A−9

S2,∞(584) = A−2x2 + A−5x − 2A−6x2 + A−6 − 2A−9x + A−10x2 + A−13x − A−14 − A−17xS2,∞(584) = −A14x2 + 3A10x2 − 2A10 + 2A7x − A6x2 − 2A3x − A2x2 + 2A2 − A−5xS2,∞(585) = A11x + A10x2 − A10 − A3x − A2x2 + A2 − A−1xS2,∞(592) = A−4x − 2A−8x + 3A−12x + A−13x2 − A−13 − A−16x − A−17x2 + A−20x + A−21 − A−24xS2,∞(592) = −A24x + 2A20x − 2A16x − A15x2 + A15 + 2A12x + A11x2 − A8x − A7 + A4x

Knots in L(5, 2)

S2,∞(22) = −A4x + x − A−4x2 + A−4 + A−8

S2,∞(23) = −A8x − x2 + x + 1 + A−4

S2,∞(31) = A−4x − A−8x2 + A−8 + A−12x2 − A−16

S2,∞(31) = A12x − A8x + A4x2 − A4 − x2 + x + A−4

S2,∞(32) = A6x2 − A2x2 + A−2x2 − A−2 − A−6x2 + A−6x + A−10

S2,∞(32) = A2x2 − A2 − A−2x2 + A−6x2 − A−10x2 + A−10x + A−14

S2,∞(34) = −A4x + 2x − A−4x2 + A−4 + A−8x2 − A−12

S2,∞(34) = A8x − A4x + x2 + x − 1 − A−4x2 + A−8

S2,∞(35) = A2x2 − 2A−2x2 + A−2 + 2A−6x2 − A−6 − A−10x2 + A−10x − A−10 + A−14

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 + A−2x2 − A−6x2 + A−6x + A−10


S2,∞(36) = A−10x2 − A−10 − A−14x2 + A−14x + A−18

S2,∞(36) = A10x2 − A10 − A−2x2 + A−2x + A−6

S2,∞(41) = −A−6x2 + A−6 + A−10x2 − 2A−14x2 + A−14x + A−14 + A−18x2 − A−18x + A−18 − A−22

S2,∞(41) = −A14x2 + A14 + A10x2 − A6x2 + A6 + A2x2 − A2x − A−2x2 + A−2x − A−2 + A−6

S2,∞(42) = −A8x + A4x2 − A4 − x2 + x + A−4x2 − A−4x − A−8x2 + A−8x − A−8

S2,∞(42) = A−4x2 − A−4 − A−8x2 + A−12x2 − A−12x − A−16x2 + A−16x − A−16

S2,∞(43) = 2A−4x − A−8x2 − A−8x + A−8 + 2A−12x2 − A−12 − A−16x2 − A−16 + A−20

S2,∞(43) = −A16x + 2A12x − A8x2 − A8x + A8 + 2A4x2 − A4 − x2 + x − 1 + A−4

S2,∞(44) = A2x2 − 3A−2x2 + 2A−2 + 2A−6x2 − 2A−10x2 + A−10x + A−14x2 − A−14x + A−14 − A−18

S2,∞(44) = −A10x2 + 2A6x2 − 3A2x2 + 2A2 + 2A−2x2 − A−2x − A−6x2 + A−6x − A−6 + A−10

S2,∞(45) = −2A4x + x2 + 2x − 1 − 2A−4x2 + A−4 + 2A−8x2 − A−8x − A−12x2 + A−12x − 2A−12

S2,∞(45) = A8x − 2A4x + 2x2 + x − 2 − 2A−4x2 + A−8x2 − A−8x + A−8 − A−12x2 + A−12x − A−12

S2,∞(47) = A−4x − A−8x + 2A−12x − A−16x2 − A−16x + A−16 + A−20x2 − A−24

S2,∞(47) = A20x − 2A16x + A12x2 + A12x − A12 − A8x2 + A4x + A4

S2,∞(48) = −A16x + A12x − A8x + A4x − x2 + 1 + A−4

S2,∞(48) = −x + A−4x − A−8x2 + A−8 + A−12x + A−12 − A−16xS2,∞(411) = −A4x + 2x − A−4x2 − A−4x + A−4 + A−8x2 − A−12x2 + A−16

S2,∞(411) = −A12x + A8x − A4x2 − A4x + A4 + x2 + x − A−4x2 + A−8

S2,∞(412) = A2x2−3A−2x2+A−2+3A−6x2−A−6−3A−10x2+A−10x+A−10+A−14x2−A−14x+2A−14−A−18

S2,∞(412) = A14 − 2A10x2 +A10 + 3A6x2 −A6 − 3A2x2 +A2 + 2A−2x2 −A−2x −A−6x2 +A−6x −A−6 +A−10

S2,∞(413) = A8x − 2A4x + x2 + 2x − 1 − 2A−4x2 + A−4 + A−8x2 + A−8 − A−12

S2,∞(414) = A8x − 3A4x + 2x2 + 2x − 2− 3A−4x2 +A−4 + 2A−8x2 −A−8x +A−8 −A−12x2 +A−12x − 2A−12

S2,∞(415) = −A6x2 + 2A2x2 −A2 − 3A−2x2 +A−2 + 3A−6x2 −A−6x −A−6 −A−10x2 +A−10x − 2A−10 +A−14

S2,∞(415) = −A10+2A6x2−A6−3A2x2+A2+2A−2x2−A−2−2A−6x2+A−6x+A−10x2−A−10x+A−10−A−14

S2,∞(416) = A−4x − A−8x2 − A−8x + A−8 + A−12x2 − A−16x2 + A−20

S2,∞(416) = −A16x + A12x − A8x2 − A8x + A8 + A4x2 − x2 + x + A−4

S2,∞(417) = A6x2 − 2A2x2 + A2 + A−2x2 − 2A−6x2 + A−6x + A−6 + A−10x2 − A−10x + A−10 − A−14

S2,∞(417) = −A6x2 + A6 + A2x2 − 2A−2x2 + A−2 + 2A−6x2 − A−6x − A−10x2 + A−10x − A−10 + A−14

S2,∞(418) = −x + A−4x2 + A−4x − A−4 − 2A−8x2 + A−8 + 2A−12x2 − A−12x − A−16x2 + A−16x − 2A−16

S2,∞(418) = A12x − 2A8x + 2A4x2 − 2A4 − 2x2 + x + A−4x2 − A−4x + A−4 − A−8x2 + A−8x − A−8

S2,∞(419) = −A10x2 + 2A6x2 − A6 − 2A2x2 + 2A−2x2 − A−2x − A−2 − A−6x2 + A−6x − A−6 + A−10

S2,∞(419) = A2x2 − A2 − 2A−2x2 + 2A−6x2 − A−6 − 2A−10x2 + A−10x + A−14x2 − A−14x + A−14 − A−18

S2,∞(420) = −A−6x2 + 2A−10x2 − 2A−10 − 2A−14x2 + A−14x + A−18x2 − A−18x + A−18 − A−22

S2,∞(420) = −A14x2 + 2A10x2 − 2A10 − A6x2 + A2x2 − A2x − A−2x2 + A−2x − A−2 + A−6

S2,∞(421) = −A4x + x2 + x − 1 − A−4x2 + A−8x2 − A−8x − A−12x2 + A−12x − A−12

S2,∞(422) = −A−12x2 + A−12x + A−12 − A−16x + A−16

S2,∞(422) = −A16x − A4x2 + A4x + A4 + 1S2,∞(423) = A−8x − A−12x2 + A−12x + A−12 + A−16x2 − A−16x − A−20

S2,∞(424) = A−16x2 − A−16x − A−16 − A−20x2 + A−20x − A−20

S2,∞(424) = A8x2 − A8x − A8 − A4 − A−4x2 + A−4xS2,∞(427) = A8x − 2A4x + x − A−4x2 − A−4x + A−4 + A−8x + A−8

S2,∞(51) = A−8x − A−12x2 + A−12x + A−12 + A−16x2 − A−16x − A−20

S2,∞(51) = A8x2 − A8 − A4x2 + A4x + 1S2,∞(52) = A−6−2A−10x2+A−10+2A−14x2−A−14−2A−18x2+A−18x+A−18+A−22x2−A−22x+A−22−A−26

S2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 − 2A10x2 + A10 + A6x2 − A6x + A6 − A2x2 + A2x − A2 + A−2


S2,∞(53) = −A12x + A8x2 + A8x − A8 − 2A4x2 + A4x + A4 + x2 + 1 − A−4

S2,∞(53) = x − A−4x + A−8x2 + A−8x − A−8 − 2A−12x2 + A−12x + A−12 + A−16x2 − A−16x + A−16 − A−20

S2,∞(54) = A14x2 − A14 − A6x2 + A6 + A2x2 − A2 − A−2x2 + A−2x + A−6

S2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 + A−14x − A−14 + A−18

S2,∞(55) = −A−12x2 + A−12 + A−16x2 − A−16x − 2A−20x2 + 2A−20x + A−24x2 − A−24x + A−24

S2,∞(55) = −A12x2 + A12 + A8x2 − A8x − A4x2 + A4x + x2 − x + 1 − A−4x2 + A−4xS2,∞(56) = A10x2 −A10 −A6x2 +A6 +A2x2 −A2x −A−2x2 +A−2x −A−2 +A−6x2 −A−6x +A−6 −A−10x2 +A−10x + A−14xS2,∞(56) = −A−6x2 + A−6 + 2A−10x2 − A−10x − A−10 − A−14x2 + A−14x − A−14 + A−18x2 − A−18x + A−18 −A−22x2 + A−22x + A−26xS2,∞(57) = −2A−6x2 + A−6 + 3A−10x2 − 2A−10 − 3A−14x2 + A−14x + 3A−18x2 − 2A−18x − A−22x2 + A−22x −2A−22 + A−26

S2,∞(57) = A18x2 − A18 − 2A14x2 + 3A10x2 − 2A10 − 3A6x2 + A6x + A6 + 2A2x2 − 2A2x + A2 − A−2x2 +A−2x − 2A−2 + A−6

S2,∞(58) = −x + A−4x2 + A−4x − A−4 − 3A−8x2 + 2A−8 + 2A−12x2 − A−12x + A−12 − 2A−16x2 + 2A−16x −A−16 + A−20x2 − A−20x + A−20

S2,∞(58) = A12x−A8x2−2A8x+A8+2A4x2−A4−3x2+2x+1+2A−4x2−2A−4x+2A−4−A−8x2+A−8x−A−8

S2,∞(59) = 2A6x2 − A6 − 3A2x2 + A2 + 3A−2x2 − A−2x − A−2 − 3A−6x2 + 2A−6x + 2A−10x2 − 2A−10x +2A−10 − A−14x2 + A−14x − A−14 + A−18xS2,∞(59) = −A6x2 + A6 + 2A2x2 − A2 − 3A−2x2 + A−2 + 4A−6x2 − 2A−6x − A−6 − 2A−10x2 + 2A−10x −2A−10 + A−14x2 − A−14x + 2A−14 − A−18x2 + A−18x + A−22xS2,∞(510) = 2A−4x − A−8x2 − 2A−8x + A−8 + 2A−12x2 + A−12x − A−12 − 2A−16x2 + A−20x2 + A−20 − A−24

S2,∞(510) = A20x − 2A16x + A12x2 + 2A12x − A12 − 2A8x2 − A8x + A8 + 2A4x2 − x2 + x − 1 + A−4

S2,∞(511) = A2x2 − 4A−2x2 + 2A−2 + 4A−6x2 −A−6 − 4A−10x2 +A−10x +A−10 + 3A−14x2 − 2A−14x +A−14 −A−18x2 + A−18x − 2A−18 + A−22

S2,∞(511) = A14x2 − 3A10x2 + A10 + 4A6x2 − A6 − 5A2x2 + A2x + 2A2 + 3A−2x2 − 2A−2x + A−2 − A−6x2 +A−6x − 2A−6 + A−10

S2,∞(512) = −2x +A−4x2 + 3A−4x −A−4 −4A−8x2 −A−8x + 3A−8 +4A−12x2 −A−12x − 3A−16x2 +2A−16x −2A−16 + A−20x2 − A−20x + 2A−20

S2,∞(512) = −A16x + 3A12x − 2A8x2 − 3A8x + 2A8 + 4A4x2 − 2A4 − 4x2 + 2x + 2A−4x2 − 2A−4x + 3A−4 −A−8x2 + A−8x − A−8

S2,∞(513) = −A6x2 + 3A2x2 − 2A2 − 4A−2x2 + A−2 + 4A−6x2 − A−6x − A−6 − 3A−10x2 + 2A−10x − A−10 +A−14x2 − A−14x + 2A−14 − A−18

S2,∞(513) = −A10x2+3A6x2−A6−4A2x2+A2+4A−2x2−A−2x−2A−2−3A−6x2+2A−6x−A−6+A−10x2−A−10x + 2A−10 − A−14

S2,∞(514) = −2A4x + x2 + 3x − 1 − 3A−4x2 − A−4x + 2A−4 + 3A−8x2 − A−8x − 3A−12x2 + 2A−12x − A−12 +A−16x2 − A−16x + 2A−16

S2,∞(514) = −A12x + 2A8x − 2A4x2 − 2A4x + 2A4 + 3x2 + x − 1 − 3A−4x2 + A−4x + 2A−8x2 − 2A−8x +2A−8 − A−12x2 + A−12x − A−12

S2,∞(515) = −A10x2 + 4A6x2 − A6 − 6A2x2 + 3A2 + 6A−2x2 − 2A−2x − 2A−2 − 4A−6x2 + 3A−6x − 2A−6 +2A−10x2 − 2A−10x + 3A−10 − A−14x2 + A−14x − A−14 + A−18xS2,∞(515) = −A6x2+4A2x2−2A2−6A−2x2+3A−2+6A−6x2−2A−6x −A−6−4A−10x2+3A−10x −2A−10+2A−14x2 − 2A−14x + 3A−14 − A−18x2 + A−18x − A−18 + A−22xS2,∞(516) = 2A8x −A4x2 − 3A4x +A4 + 3x2 + 2x − 2− 4A−4x2 +A−4x +A−4 + 3A−8x2 − 2A−8x + 2A−8 −A−12x2 + A−12x − 2A−12


S2,∞(516) = A8x − 3A4x + 2x2 + 3x − 2 − 4A−4x2 + 2A−4 + 3A−8x2 − A−8x + A−8 − 2A−12x2 + 2A−12x −2A−12 + A−16x2 − A−16x + A−16

S2,∞(517) = A14x2 − 2A10x2 + 4A6x2 − 2A6 − 4A2x2 +A2x +A2 + 2A−2x2 −A−2x +A−2 −A−6x2 +A−6x −A−6 + A−10

S2,∞(517) = A2x2−3A−2x2+A−2+4A−6x2−2A−6−3A−10x2+A−10x +2A−14x2−A−14x +A−14−A−18x2+A−18x − A−18 + A−22

S2,∞(518) = −2A−6x2 + A−6 + 3A−10x2 − A−10 − 4A−14x2 + A−14x + 2A−14 + 3A−18x2 − 2A−18x + A−18 −A−22x2 + A−22x − 2A−22 + A−26

S2,∞(518) = A18x2 − 3A14x2 + 2A14 + 3A10x2 − A10 − 3A6x2 + A6x + A6 + 2A2x2 − 2A2x + A2 − A−2x2 +A−2x − 2A−2 + A−6

S2,∞(519) = −2A4x + x2 + 2x − 1 − 3A−4x2 + 2A−4 + 2A−8x2 − A−8x + A−8 − 2A−12x2 + 2A−12x − A−12 +A−16x2 − A−16x + A−16

S2,∞(519) = A8x − A4x2 − 2A4x + A4 + 2x2 + x − 1 − 3A−4x2 + A−4x + A−4 + 2A−8x2 − 2A−8x + 2A−8 −A−12x2 + A−12x − A−12

S2,∞(520) = 2A2x2 −A2 − 4A−2x2 + 2A−2 + 4A−6x2 −A−6x −A−6 − 3A−10x2 + 2A−10x −A−10 + 2A−14x2 −2A−14x + 2A−14 − A−18x2 + A−18x − A−18 + A−22xS2,∞(520) = −A10x2 + 3A6x2 − A6 − 4A2x2 + 2A2 + 4A−2x2 − 2A−2x − A−2 − 2A−6x2 + 2A−6x − 2A−6 +A−10x2 − A−10x + 2A−10 − A−14x2 + A−14x + A−18xS2,∞(521) = 2A12x−A8x2−2A8x+A8+3A4x2−2A4−3x2+2x+2A−4x2−2A−4x+2A−4−A−8x2+A−8x−A−8

S2,∞(521) = −x +A−4x2 + 2A−4x −A−4 − 3A−8x2 + 2A−8 + 3A−12x2 −A−12x − 2A−16x2 + 2A−16x − 2A−16 +A−20x2 − A−20x + A−20

S2,∞(522) = −A6x2 + 2A2x2 −A2 − 3A−2x2 + 2A−2 + 2A−6x2 −A−6x +A−6 −A−10x2 +A−10x −A−10 +A−14

S2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 + 2A−2x2 − A−2 − 2A−6x2 + A−6x + A−10x2 − A−10x + A−10 − A−14

S2,∞(523) = −A12x + A8x − A4x2 + A4 + x2 + x − A−4

S2,∞(523) = 2x − A−4x + A−8x2 − A−8 − A−12x2 + A−16

S2,∞(524) = A−6 + A−14x − A−14

S2,∞(524) = −A18 + A14x2 − A14 + A6 − A−2x2 + A−2x + A−6

S2,∞(525) = −A−6x2+2A−10x2−4A−14x2+2A−14+4A−18x2−A−18−3A−22x2+A−22x +A−26x2−A−26x +2A−26 − A−30

S2,∞(525) = A26 − 2A22x2 + 4A18x2 − A18 − 4A14x2 + 2A14 + 3A10x2 − A10x − 2A6x2 + A6x − A6 + A2

S2,∞(526) = −x + 3A−4x −A−8x2 − 2A−8x +A−8 + 2A−12x2 + 2A−12x −A−12 − 2A−16x2 −A−16x +A−20x2 +A−20 − A−24

S2,∞(526) = A20x − 3A16x + A12x2 + 3A12x − A12 − 2A8x2 − A8x + A8 + 2A4x2 + A4x − x2 − 1 + A−4

S2,∞(527) = −x + 3A−4x −A−8x2 − 2A−8x +A−8 + 2A−12x2 + 2A−12x −A−12 − 2A−16x2 −A−16x +A−20x2 +A−20 − A−24

S2,∞(527) = A20x − 3A16x + A12x2 + 3A12x − A12 − 2A8x2 − A8x + A8 + 2A4x2 + A4x − x2 − 1 + A−4

S2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 + 2A−2x2 − A−6x2 + A−6x − A−6 + A−10

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 + 3A−6x2 − A−6 − 3A−10x2 + A−10x + A−14x2 + A−14

S2,∞(530) = A8x − A4x + 2x − 2A−4x + A−8x2 + A−8x − A−8 − A−12x2 + A−16

S2,∞(530) = −A12x + 2A8x − A4x2 − A4x + A4 + x2 + x − A−4x − A−4 + A−8xS2,∞(531) = A8x − 2A4x + 3x − A−4x2 − 2A−4x + A−4 + 2A−8x2 + A−8x − A−8 − A−12x2 − A−12 + A−16

S2,∞(531) = −A12x + 3A8x − A4x2 − 2A4x + A4 + 2x2 + x − 1 − A−4x2 − A−4x − A−4 + A−8x + A−8

S2,∞(532) = A8x − 2A4x + 2x − A−4x2 − 2A−4x + A−4 + A−8x2 + A−8x − A−12x2 + A−16

S2,∞(532) = −A12x + 2A8x − A4x2 − 2A4x + A4 + x2 + x − A−4x2 − A−4x + A−8x + A−8

S2,∞(534) = −A6x2 + 3A2x2 − A2 − 3A−2x2 + A−2 + 3A−6x2 − A−6 − 2A−10x2 + A−10x − A−10 + 2A−14


S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 + 3A−2x2 − A−2 − 2A−6x2 + A−6x + A−10

S2,∞(535) = −A−6x2+2A−10x2−A−10−3A−14x2+4A−18x2−2A−18−3A−22x2+A−22x +A−26x2−A−26x +2A−26 − A−30

S2,∞(535) = A26 − 2A22x2 + 4A18x2 − 2A18 − 3A14x2 + 3A10x2 − A10x − A10 − 2A6x2 + A6x − A6 + A2

S2,∞(536) = −x + 2A−4x − A−8x2 − 2A−8x + A−8 + A−12x2 + 2A−12x − 2A−16x2 − A−16x + A−16 + A−20x2 +A−20 − A−24

S2,∞(536) = A20x − 3A16x + A12x2 + 2A12x − A12 − 2A8x2 − A8x + A8 + A4x2 + A4x + A4 − x2 + A−4

S2,∞(538) = −A−2+A−6x2−A−6−2A−10x2+2A−14x2−A−14−2A−18x2+A−18x +A−18+A−22x2−A−22x +A−22 − A−26

S2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 − 2A10x2 + 2A6x2 − A6x − A6 − A2x2 + A2x − 2A2 + A−2

S2,∞(539) = −A12x + A8x2 + A8x − A8 − 2A4x2 + A4 + x2 + 1 − A−4x2 + A−8

S2,∞(539) = −A4x + x −A−4x2 −A−4x +A−4 +A−8x2 +A−8x −2A−12x2 +A−12x +A−12 +A−16x2 −A−16x +A−16 − A−20

S2,∞(540) = A−8x − A−12x2 + A−12 + A−16x2 − A−20x2 + A−24x2 − A−28

S2,∞(540) = A24x − A20x + A16x2 − A16 − A12x2 + A8x2 − A4x2 + A4x + 1S2,∞(541) = A18 + A10x2 − A10 − A6x2 + A2x2 − A2 − A−2x2 + A−2x + A−6

S2,∞(541) = A−2x2 − A−2 − A−6x2 + A−10x2 − A−10 − A−14x2 + A−14x + 2A−18

S2,∞(542) = A−4x − A−8x2 − A−8x + A−8 + A−12x2 + A−12x − A−16x2 + A−20x2 − A−24

S2,∞(542) = A20x − A16x + A12x2 + A12x − A12 − A8x2 − A8x + A4x2 − x2 + x + A−4

S2,∞(543) = A6x2 − 2A2x2 + A2 + 2A−2x2 − A−2 − 2A−6x2 + A−6x + 2A−10x2 − A−10x − A−14x2 + A−14x −A−14 + A−18

S2,∞(543) = A10x2 − A10 − A6x2 + 2A2x2 − A2 − 3A−2x2 + A−2x + A−2 + 2A−6x2 − A−6x + A−6 − A−10x2 +A−10x − A−10 + A−14

S2,∞(544) = −x +A−4x2 + 2A−4x −A−4 − 3A−8x2 −A−8x + 2A−8 + 3A−12x2 −A−12x − 3A−16x2 + 2A−16x −A−16 + A−20x2 − A−20x + 2A−20

S2,∞(544) = −A16x + 2A12x − 2A8x2 − 2A8x + 2A8 + 3A4x2 − A4 − 3x2 + 2x + 2A−4x2 − 2A−4x + 2A−4 −A−8x2 + A−8x − A−8

S2,∞(545) = −A10x2 + 3A6x2 −A6 − 4A2x2 + 2A2 + 3A−2x2 −A−2x − 3A−6x2 + 2A−6x +A−10x2 −A−10x +2A−10 − A−14

S2,∞(545) = −A6x2 + A6 + 2A2x2 − 4A−2x2 + 2A−2 + 4A−6x2 − A−6x − A−6 − 3A−10x2 + 2A−10x − A−10 +A−14x2 − A−14x + 2A−14 − A−18

S2,∞(546) = −x + A−4x2 + 3A−4x − A−4 − 3A−8x2 − A−8x + 2A−8 + 4A−12x2 − A−12x − A−12 − 3A−16x2 +2A−16x − 2A−16 + A−20x2 − A−20x + 2A−20

S2,∞(546) = −A16x +3A12x −2A8x2−2A8x +2A8+4A4x2−2A4−3x2+2x − 1+2A−4x2−2A−4x +2A−4−A−8x2 + A−8x − A−8

S2,∞(547) = −A10x2 + 4A6x2 − 2A6 − 5A2x2 + 2A2 + 5A−2x2 − 2A−2x − A−2 − 4A−6x2 + 3A−6x − A−6 +2A−10x2 − 2A−10x + 3A−10 − A−14x2 + A−14x − A−14 + A−18xS2,∞(547) = −A6x2 + A6 + 3A2x2 − A2 − 5A−2x2 + 2A−2 + 6A−6x2 − 2A−6x − 2A−6 − 4A−10x2 + 3A−10x −2A−10 + 2A−14x2 − 2A−14x + 3A−14 − A−18x2 + A−18x − A−18 + A−22xS2,∞(548) = A4x − x2 − x + 1+ 2A−4x2 +A−4x −A−4 − 3A−8x2 +A−8x +A−8 + 3A−12x2 − 2A−12x +A−12 −A−16x2 + A−16x − 2A−16

S2,∞(548) = A12x − 2A8x + 2A4x2 + A4x − 2A4 − 3x2 + x + 1+ 2A−4x2 − A−4x + A−4 − 2A−8x2 + 2A−8x −A−8 + A−12x2 − A−12x + A−12

S2,∞(549) = A14x2−2A10x2+A10+3A6x2−A6−3A2x2+A2x+2A−2x2−A−2x−A−6x2+A−6x−A−6+A−10

S2,∞(549) = A2x2 − A2 − 2A−2x2 + 3A−6x2 − A−6 − 3A−10x2 + A−10x + A−10 + 2A−14x2 − A−14x + A−14 −


A−18x2 + A−18x − A−18 + A−22

S2,∞(551) = A−4x − A−8x + 3A−12x − A−16x2 − 2A−16x + A−16 + 2A−20x2 − A−20 − A−24x2 − A−24 + A−28

S2,∞(551) = −A24x + 2A20x − A16x2 − 2A16x + A16 + 2A12x2 + A12x − A12 − A8x2 − A8 + A4x + A4

S2,∞(552) = −A16x + A12x − A8x + 2A4x − x2 + 1 + A−4x2 − A−8

S2,∞(552) = A4x − x + A−4x2 + A−4x − A−4 − A−8x2 + A−12x + A−12 − A−16xS2,∞(553) = A−2x2 − 2A−6x2 + A−6 + 2A−10x2 − A−10 − 2A−14x2 + A−14x + A−18x2 + A−18 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 + A2x2 − A−2x2 + A−2x + A−6

S2,∞(554) = A20x − 2A16x + A12x2 + 2A12x − A12 − A8x2 − A8x + A4x2 + A4x − x2 + A−4

S2,∞(554) = −x + 2A−4x − A−8x2 − A−8x + A−8 + A−12x2 + 2A−12x − A−16x2 − A−16x + A−20x2 − A−24

S2,∞(555) = −A18x2 + 3A14x2 − A14 − 4A10x2 + 2A10 + 3A6x2 − A6x − 2A2x2 + A2x − A2 + 2A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − 4A−10x2 + 2A−10 + 3A−14x2 − A−14 − 2A−18x2 + A−18x + A−22x2 −A−22x + A−22 − A−26

S2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 + 2A−2x2 − A−2 − A−6x2 + A−6x − A−6 + A−10

S2,∞(556) = −A6 + 2A2x2 − A2 − 2A−2x2 + 2A−6x2 − 3A−10x2 + A−10x + A−10 + A−14x2 + A−14

S2,∞(558) = −A4x + 2x − A−4x2 − A−4x + A−4 + A−8x2 + A−8x − A−12x2 + A−16x2 − A−20

S2,∞(558) = A16x − A12x + A8x2 + A8x − A8 − A4x2 − A4x + x2 + x − A−4x2 + A−8

S2,∞(559) = A2x2 − 3A−2x2 +A−2 +4A−6x2 −A−6 −4A−10x2 +A−10x +A−10 + 3A−14x2 −A−14x −A−18x2 +A−18x − 2A−18 + A−22

S2,∞(559) = −A18 + 2A14x2 − A14 − 3A10x2 + A10 + 4A6x2 − A6 − 4A2x2 + A2x + A2 + 2A−2x2 − A−2x +A−2 − A−6x2 + A−6x − A−6 + A−10

S2,∞(560) = A8x − 2A4x + x2 + 3x − 1 − 2A−4x2 − A−4x + A−4 + 2A−8x2 − A−12x2 − A−12 + A−16

S2,∞(560) = −A12x + 2A8x − A4x2 − 2A4x + A4 + 2x2 + 2x − 1 − 2A−4x2 + A−8x2 + A−8 − A−12

S2,∞(561) = A2x2−4A−2x2+A−2+5A−6x2−2A−6−5A−10x2+A−10x+A−10+4A−14x2−2A−14x−A−18x2+A−18x − 3A−18 + A−22

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 + 5A6x2 − 2A6 − 5A2x2 + A2x + A2 + 3A−2x2 − 2A−2x +A−2 − A−6x2 + A−6x − 2A−6 + A−10

S2,∞(562) = A8x − 4A4x + 2x2 + 4x − 2 − 5A−4x2 − A−4x + 3A−4 + 4A−8x2 − A−8x + A−8 − 3A−12x2 +2A−12x − 2A−12 + A−16x2 − A−16x + 2A−16

S2,∞(562) = −A12x + 3A8x − 2A4x2 − 4A4x + 2A4 + 4x2 + 2x − 2 − 5A−4x2 + A−4x + A−4 + 3A−8x2 −2A−8x + 3A−8 − A−12x2 + A−12x − 2A−12

S2,∞(563) = −A6x2 + 3A2x2 −A2 − 5A−2x2 + 2A−2 + 5A−6x2 −A−6x −A−6 − 4A−10x2 + 2A−10x +A−14x2 −A−14x + 3A−14 − A−18

S2,∞(563) = A14 − 2A10x2 + A10 + 4A6x2 − A6 − 5A2x2 + 2A2 + 4A−2x2 − A−2x − A−2 − 3A−6x2 + 2A−6x −A−6 + A−10x2 − A−10x + 2A−10 − A−14

S2,∞(564) = A8x − 3A4x + 2x2 + 4x − 2 − 4A−4x2 − A−4x + 2A−4 + 4A−8x2 − A−8x − 3A−12x2 + 2A−12x −2A−12 + A−16x2 − A−16x + 2A−16

S2,∞(564) = −A12x + 3A8x − 2A4x2 − 3A4x + 2A4 + 4x2 + 2x − 2 − 4A−4x2 + A−4x + 3A−8x2 − 2A−8x +2A−8 − A−12x2 + A−12x − 2A−12

S2,∞(565) = −A6x2+5A2x2−2A2−8A−2x2+3A−2+9A−6x2−2A−6x −3A−6−7A−10x2+4A−10x −A−10+3A−14x2 − 3A−14x + 5A−14 − A−18x2 + A−18x − 2A−18 + A−22xS2,∞(565) = A14−3A10x2+A10+7A6x2−3A6−8A2x2+3A2+8A−2x2−3A−2x−2A−2−5A−6x2+4A−6x−3A−6 + 2A−10x2 − 2A−10x + 4A−10 − A−14x2 + A−14x − A−14 + A−18xS2,∞(566) = A10x2−2A6x2+A6+3A2x2−A2−4A−2x2+A−2x +A−2+3A−6x2−A−6x −A−10x2+A−10x −2A−10 + A−14

S2,∞(566) = −A10 + 2A6x2 −A6 − 3A2x2 +A2 + 3A−2x2 −A−2 − 3A−6x2 +A−6x +A−6 + 2A−10x2 −A−10x +


A−10 − A−14x2 + A−14x − A−14 + A−18

S2,∞(567) = −A−6x2 + A−6 + 2A−10x2 − A−10 − 2A−14x2 + A−14x + 2A−18x2 − A−18x − A−22x2 + A−22x −A−22 + A−26

S2,∞(567) = A18x2−A18−A14x2+2A10x2−A10−2A6x2+A6x+A6+A2x2−A2x+A2−A−2x2+A−2x−A−2+A−6

S2,∞(568) = −A8x+A4x2−A4−2x2+x+1+A−4x2−A−4x+A−4−2A−8x2+2A−8x+A−12x2−A−12x+A−12

S2,∞(568) = −x2 + 1 + A−4x2 − 2A−8x2 + A−8x + A−8 + 2A−12x2 − 2A−12x + A−12 − A−16x2 + A−16x − A−16

S2,∞(569) = A−2x2−A−2−2A−6x2+A−6+3A−10x2−A−10x−A−10−3A−14x2+2A−14x+2A−18x2−2A−18x+2A−18 − A−22x2 + A−22x − A−22 + A−26xS2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 − 2A6x2 + A6 + 3A2x2 − 2A2x − A2 − 2A−2x2 + 2A−2x − 2A−2 +A−6x2 − A−6x + 2A−6 − A−10x2 + A−10x + A−14xS2,∞(570) = A12x−A8x2−A8x+A8+2A4x2−A4−2x2+2x+2A−4x2−2A−4x+A−4−A−8x2+A−8x−A−8

S2,∞(570) = A−4x2 +A−4x −A−4 −2A−8x2 +A−8 +2A−12x2 −A−12x −2A−16x2 +2A−16x −A−16 +A−20x2 −A−20x + A−20

S2,∞(571) = −A6x2 + A6 + A2x2 − A2x − A2 + A−2x − 2A−2

S2,∞(571) = −A6 + A2x2 − A2 − A−2 − A−6x2 + A−6 − A−14x2 + A−14x + A−18x2 − A−18x + A−18 − A−22

S2,∞(572) = A−8x−A−12x2+A−12+2A−16x2−A−16x−A−16−2A−20x2+2A−20x−A−20+A−24x2−A−24x+A−24

S2,∞(572) = A16x − A12x2 + A12 + 2A8x2 − A8x − A8 − A4x2 + A4x − A4 + x2 − x + 1 − A−4x2 + A−4xS2,∞(573) = A6x2 − 2A2x2 +A2 + 2A−2x2 −A−2x −A−2 −A−6x2 +A−6x −A−6 +A−10x2 −A−10x +A−10 −A−14x2 + A−14x + A−18xS2,∞(573) = A2x2 − A2 − 2A−2x2 + A−2 + 2A−6x2 − A−6x − A−10x2 + A−10x − A−10 + A−14x2 − A−14x +A−14 − A−18x2 + A−18x + A−22xS2,∞(574) = A−18xS2,∞(574) = −A22 + A18x2 − A2x2 + A2x + A−2

S2,∞(575) = −A6x2 + A6 + A2x2 − A2x − A−2x2 + A−2x + A−6

S2,∞(575) = A2 − A−6x2 + A−6 − A−14x2 + A−14x + A−18x2 − A−18x + A−18 − A−22

S2,∞(576) = −x2 + 1 + A−4x2 − A−4x − A−8x2 + A−8x + A−12

S2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 + A−6x2 − A−10x2 + A−10x − A−10 + A−14

S2,∞(577) = −A10 + A6x2 − A2x2 + 2A−2x2 − A−2 − 2A−6x2 + A−6x + A−6 + A−10

S2,∞(578) = −x2 + x + 1 + A−4x2 − A−4x + A−8x − A−8

S2,∞(578) = A8x + A−4x2 − A−4x − A−4 − A−8x2 + A−8x + A−12

S2,∞(579) = −A−6x2 + A−6 + A−10x2 − 2A−14x2 + A−14 + 2A−18x2 − 2A−22x2 + A−22x + A−26x2 − A−26x +A−26 − A−30

S2,∞(579) = −A22x2 + 2A18x2 − 2A14x2 + A14 + 2A10x2 − A10x − 2A6x2 + A6x + A2

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 + A−2x2 − A−6x2 + A−6x + A−10

S2,∞(580) = A2x2 − A−2x2 + A−6x2 − 2A−10x2 + A−10x + A−14x2

S2,∞(581) = −A−14 + A−18x2 + A−18x − A−18 − A−22x2 + A−22 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A2x2 + A2x + A−2

S2,∞(582) = −A−6x2 + 3A−10x2 − 2A−10 − 3A−14x2 + A−14x + A−14 + 2A−18x2 − A−18x + A−18 − A−22x2 +A−22x − A−22 + A−26

S2,∞(582) = A18x2−2A14x2+A14+3A10x2−2A10−2A6x2+A6x+A2x2−A2x+A2−A−2x2+A−2x−A−2+A−6

S2,∞(583) = −A4x + x2 + 2x − 1 − 2A−4x2 + A−4 + 2A−8x2 − A−8x − 2A−12x2 + 2A−12x − A−12 + A−16x2 −A−16x + A−16

S2,∞(583) = A8x−A4x2−A4x+A4+2x2+x−1−2A−4x2+A−4x+2A−8x2−2A−8x+A−8−A−12x2+A−12x−A−12

S2,∞(584) = A−2x2 − 3A−6x2 + 2A−6 + 4A−10x2 − A−10x − 2A−10 − 3A−14x2 + 2A−14x − A−14 + 2A−18x2 −2A−18x + 2A−18 − A−22x2 + A−22x − A−22 + A−26x


S2,∞(584) = −A14x2 + 3A10x2 − 2A10 − 3A6x2 + 2A6 + 3A2x2 − 2A2x − 2A−2x2 + 2A−2x − 2A−2 + A−6x2 −A−6x + 2A−6 − A−10x2 + A−10x + A−14xS2,∞(585) = A−22x2 − A−22x − A−26x2 + A−26x + A−30xS2,∞(585) = A6x2 − A6x − A2 − A−6x2 + A−6x + A−6 + A−10xS2,∞(592) = A−4x − 2A−8x + 3A−12x − A−16x2 − A−16x + A−16 + A−20x2 + A−20x − A−24x − A−24

S2,∞(592) = −A24x + 2A20x − 2A16x + A12x2 + 2A12x − A12 − A8x2 − A8x + A4x + A4

Knots in L(6, 1)

S2,∞(32) = A6x2 − A2x2 − x2 + 1 + A−4

S2,∞(32) = A2x2 − A2 − A−2x2 − A−4x2 + A−4 + A−6 + A−8

S2,∞(35) = A2x2 − 2A−2x2 + A−2 − A−4x2 + A−4 + A−6x2 + A−8 − A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 − x2 + 1 + A−2 + A−4

S2,∞(36) = A10x2 − A10 − A4x2 + A4 − A2x2 + A2 + 1S2,∞(41) = −A−6x2 + A−6 − A−8x2 + A−8 + A−10 + A−12x2 − A−16

S2,∞(41) = −A14x2 + A14 + A10x2 + A8x2 − A8 − A4x2 − A2x2 + A2 + 1S2,∞(42) = −A8x3 + A8x + A4x3 − A4x + xS2,∞(42) = −x3 + 2x + A−4x3 − A−4xS2,∞(44) = A2x2 − 3A−2x2 + 2A−2 − A−4x2 + A−4 + A−6x2 + A−6 + A−8x2 − A−10 − A−12

S2,∞(44) = −A10x2 + 2A6x2 + A4x2 − A4 − 2A2x2 + A2 − x2 + A−2 + A−4

S2,∞(45) = −A4x3 + 2x3 − x − A−4x3 + A−4x + A−8xS2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x − A−4xS2,∞(412) = A2x2 − 3A−2x2 + A−2 − A−4x2 + A−4 + 2A−6x2 + A−8x2 − A−10x2 − A−12 + A−14

S2,∞(412) = A14 − 2A10x2 + A10 + 3A6x2 − A6 + A4x2 − A4 − 2A2x2 − x2 + A−2 + A−4

S2,∞(414) = A8x − 2A4x3 + A4x + 3x3 − 2x − A−4x3 + A−8xS2,∞(415) = −A6x2 + 2A2x2 − A2 + x2 − 1 − 2A−2x2 − A−4x2 + A−6x2 + A−8 − A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − x2 + 1 + A−2x2 + A−4x2 − A−6 − A−8

S2,∞(417) = A6x2 − 2A2x2 + A2 − x2 + 1 + A−2 + A−4x2 − A−8

S2,∞(417) = −A6x2 + A6 + A2x2 + x2 − 1 − A−2x2 − A−4x2 + A−6 + A−8

S2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x − A−8x3 + A−8x + A−12xS2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4xS2,∞(419) = −A10x2 + 2A6x2 − A6 + A4x2 − A4 − A2x2 − A2 − x2 + A−4

S2,∞(419) = A2x2 − A2 − 2A−2x2 − A−4x2 + A−4 + A−6x2 + A−8x2 − A−10 − A−12

S2,∞(420) = −A−6x2 − A−8x2 + A−8 + A−10x2 − A−10 + A−12x2 − A−14 − A−16

S2,∞(420) = −A14x2 + 2A10x2 − 2A10 + A8x2 − A8 − A6 − A4x2 − A2x2 + A2 + 1S2,∞(421) = −A4x3 + A4x + x3

S2,∞(424) = A−12xS2,∞(424) = −A12x3 + 2A12x + A4x3 − A4xS2,∞(52) = A−6 − 2A−10x2 + A−10 − A−12x2 + A−12 + A−14x2 + A−16x2 − A−20

S2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 + A12x2 − A12 − A10x2 − A8x2 − A6x2 + 2A6 + A4

S2,∞(54) = A14x2 − A14 − A6x2 + A6 − A4x2 + A4 + 1S2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 − A−8x2 + A−8 − A−10x2 + A−10 + A−12 + A−14x2 − A−14

S2,∞(55) = A16x3 − 2A16x − A12x3 + A12x − A8x + A4x3 − A4x


S2,∞(56) = A10x2 − A10 + A8x2 − A8 − A4x2 − A2x2 + A2 − A−4

S2,∞(57) = −2A−6x2+A−6−A−8x2+A−8+2A−10x2−A−10+2A−12x2−A−12−2A−14−A−16x2−A−16+A−20

S2,∞(57) = A18x2−A18−2A14x2−A12x2+A12+2A10x2−A10+2A8x2−A8−A6−A4x2−A4−A2x2+A2+ 1S2,∞(58) = −x3 + x + 3A−4x3 − 4A−4x − A−8x3

S2,∞(58) = A12x3 − A12x − 2A8x3 + A8x + 2A4x3 − 3A4xS2,∞(59) = 2A6x2 − A6 + A4x2 − A4 − 2A2x2 − 2x2 + 1 + A−2 + A−4x2 − 2A−8

S2,∞(59) = −A6x2 + A6 + 2A2x2 − A2 + 2x2 − 2 − A−2x2 − A−2 − 2A−4x2 + A−6 + A−8 − A−12

S2,∞(511) = A2x2 − 4A−2x2 + 2A−2 − A−4x2 + A−4 + 3A−6x2 + 2A−8x2 − A−8 − A−10x2 − A−10 − A−12x2 −A−12 + A−14 + A−16

S2,∞(511) = A14x2 − 3A10x2 + A10 − A8x2 + A8 + 3A6x2 + 2A4x2 − A4 − 2A2x2 − x2 − 1 + A−2 + A−4

S2,∞(512) = −x3 + 4A−4x3 − 4A−4x − 3A−8x3 + 2A−8x + A−12x3 − A−16xS2,∞(512) = −A16x + 2A12x3 − A12x − 4A8x3 + 3A8x + 3A4x3 − 3A4x − xS2,∞(513) = −A6x2+3A2x2−2A2+x2−1−3A−2x2−2A−4x2+A−4+A−6x2+A−6+A−8x2+A−8−A−10−A−12

S2,∞(513) = −A10x2 + 3A6x2 − A6 + A4x2 − A4 − 3A2x2 − 2x2 + 1 + A−2x2 + A−4x2 + A−4 − A−6 − A−8

S2,∞(514) = −A4x3 + 3x3 − 2x − 2A−4x3 + A−4x + A−8x3 − A−8x − A−12xS2,∞(514) = −A12x + 2A8x3 − 2A8x − 3A4x3 + 2A4x + 2x3 − x − A−4xS2,∞(515) = −A10x2+4A6x2−A6+2A4x2−2A4−4A2x2+A2−3x2+1+A−2x2+A−2+A−4x2+A−4−A−6−2A−8

S2,∞(515) = −A6x2 + 4A2x2 − 2A2 + 2x2 − 2 − 4A−2x2 + A−2 − 3A−4x2 + A−4 + A−6x2 + 2A−6 + A−8x2 +A−8 − A−10 − 2A−12

S2,∞(516) = A8x3 − 3A4x3 + 2A4x + 3x3 − 2x − A−4x3 + A−8xS2,∞(516) = A8x − 2A4x3 + A4x + 4x3 − 3x − 2A−4x3 + A−4x + A−8xS2,∞(517) = A14x2 − 2A10x2 − A8x2 + A8 + 3A6x2 − A6 + A4x2 − 2A2x2 − x2 + A−2 + A−4

S2,∞(517) = A2x2−3A−2x2+A−2−A−4x2+A−4+3A−6x2−A−6+A−8x2−A−10x2−A−10−A−12x2+A−14+A−16

S2,∞(518) = −2A−6x2+A−6−A−8x2+A−8+2A−10x2+2A−12x2−A−12−A−14x2−A−16x2−A−16+A−18+A−20

S2,∞(518) = A18x2 − 3A14x2 + 2A14 −A12x2 +A12 + 2A10x2 + 2A8x2 −A8 −A6 −A4x2 −A4 −A2x2 +A2 + 1S2,∞(519) = −A4x3 + 3x3 − 3x − A−4x3

S2,∞(519) = A8x3 − A8x − 2A4x3 + A4x + 2x3 − 2x − A−4xS2,∞(520) = 2A2x2 − A2 + x2 − 1 − 3A−2x2 + A−2 − 2A−4x2 + A−4 + A−6x2 + A−6 + A−8x2 − A−10 − 2A−12

S2,∞(520) = −A10x2 + 3A6x2 − A6 + 2A4x2 − 2A4 − 2A2x2 − 2x2 + A−2 + A−4 − A−8

S2,∞(521) = A12x3 − 3A8x3 + 3A8x + 2A4x3 − 2A4xS2,∞(521) = −x3 + x + 3A−4x3 − 3A−4x − 2A−8x3 + 2A−8x + A−12xS2,∞(522) = −A6x2 + 2A2x2 − A2 + x2 − 1 − 2A−2x2 + A−2 − A−4x2 + 2A−6 + A−8

S2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 − x2 + 1 + A−2x2 + A−4x2 − A−6 − A−8

S2,∞(524) = A−6 − A−8x2 + A−8 − A−10x2 + A−10 + A−12 + A−14x2 − A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 + A6 − A4x2 + A4 − A2x2 + A2 + 1S2,∞(525) = −A−6x2 + 2A−10x2 − 4A−14x2 + 2A−14 −A−16x2 +A−16 + 3A−18x2 +A−20x2 −A−22x2 −A−22 −A−24 + A−26

S2,∞(525) = A26 − 2A22x2 + 4A18x2 − A18 + A16x2 − A16 − 3A14x2 + A14 − A12x2 + A10x2 + A10 + A8 − A6x2

S2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 − x2 + 1 + A−2x2 + A−2 + A−4 − A−6

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 − A−4x2 + A−4 + 2A−6x2 + A−8 − 2A−10x2 + A−14x2

S2,∞(534) = −A6x2 + 3A2x2 − A2 − 3A−2x2 + A−2 − A−4x2 + A−4 + 2A−6x2 + A−8 − A−10x2 − A−10 + A−14

S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 − x2 + 1 + 2A−2x2 + A−4 − A−6x2

S2,∞(535) = −A−6x2 + 2A−10x2 − A−10 − 3A−14x2 − A−16x2 + A−16 + 3A−18x2 − A−18 + A−20x2 − A−22x2 −A−22 − A−24 + A−26

S2,∞(535) = A26 − 2A22x2 + 4A18x2 − 2A18 + A16x2 − A16 − 2A14x2 − A14 − A12x2 + A10x2 + A8 − A6x2


S2,∞(538) = −A−2 + A−6x2 − A−6 − 2A−10x2 − A−12x2 + A−12 + A−14x2 + A−16x2 − A−20

S2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 + A12x2 − A12 − A10x2 − A10 − A8x2 + A4 − A2

S2,∞(541) = A18 + A10x2 − A10 − A6x2 − A4x2 + A4 + 1S2,∞(541) = A−2x2 − A−2 − A−6x2 − A−8x2 + A−8 + A−12 + A−18

S2,∞(543) = A6x2 − 2A2x2 + A2 − x2 + 1 + A−2x2 + A−4x2 − A−6 − A−8x2 + A−12

S2,∞(543) = A10x2 − A10 − A6x2 − A4x2 + A4 + A2x2 + x2 − A−2x2 − A−4x2 + A−6 + A−8

S2,∞(544) = −x3 + x + 3A−4x3 − 3A−4x − 2A−8x3 + A−8x + A−12x3 − A−12x − A−16xS2,∞(544) = −A16x + 2A12x3 − 2A12x − 3A8x3 + 2A8x + 2A4x3 − 2A4xS2,∞(545) = −A10x2 + 3A6x2 − A6 + A4x2 − A4 − 3A2x2 + A2 − 2x2 + 1 + 2A−2 + A−4x2 + A−4 − A−8

S2,∞(545) = −A6x2 +A6 + 2A2x2 + x2 − 1− 3A−2x2 +A−2 − 2A−4x2 +A−4 +A−6x2 +A−6 +A−8x2 +A−8 −A−10 − A−12

S2,∞(546) = −x3 + x + 3A−4x3 − 2A−4x − 3A−8x3 + 3A−8x + A−12x3 − A−16xS2,∞(546) = −A16x + 2A12x3 − A12x − 4A8x3 + 4A8x + 2A4x3 − A4xS2,∞(547) = −A10x2 + 4A6x2 − 2A6 + 2A4x2 − 2A4 − 3A2x2 − 3x2 + 1 + 2A−2 + A−4x2 + A−4 − 2A−8

S2,∞(547) = −A6x2 +A6 + 3A2x2 −A2 + 2x2 − 2− 3A−2x2 − 3A−4x2 +A−4 +A−6x2 +A−6 +A−8x2 +A−8 −A−10 − 2A−12

S2,∞(548) = A4x3 − A4x − 2x3 + 2x + 2A−4x3 − 2A−4x − A−8x3 + A−8x + A−12xS2,∞(548) = A12x − 2A8x3 + 2A8x + 3A4x3 − 3A4x − x3 + xS2,∞(549) = A14x2 − 2A10x2 + A10 − A8x2 + A8 + 2A6x2 + A4x2 − A2x2 − A2 − x2 + A−4

S2,∞(549) = A2x2 − A2 − 2A−2x2 − A−4x2 + A−4 + 2A−6x2 + A−8x2 − A−10x2 − A−12x2 + A−14 + A−16

S2,∞(553) = A−2x2 − 2A−6x2 + A−6 − A−8x2 + A−8 + A−10x2 + A−12 − A−14x2 + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 − A4x2 + A4 + A2 + 1S2,∞(555) = −A18x2 + 3A14x2 − A14 + A12x2 − A12 − 3A10x2 + A10 − A8x2 + A6x2 + A6 + A4 − A2x2 + A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − 4A−10x2 + 2A−10 − A−12x2 + A−12 + 2A−14x2 + A−16x2 − A−18 − A−20

S2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 − x2 + 1 + A−2x2 + A−4 − A−6

S2,∞(556) = −A6 + 2A2x2 − A2 − 2A−2x2 − A−4x2 + A−4 + A−6x2 + A−6 + A−8 − 2A−10x2 + A−10 + A−14x2

S2,∞(559) = A2x2−3A−2x2+A−2−A−4x2+A−4+3A−6x2+A−8x2−2A−10x2−A−12x2+A−14x2+A−16−A−18

S2,∞(559) = −A18 + 2A14x2 − A14 − 3A10x2 + A10 − A8x2 + A8 + 3A6x2 + A4x2 − 2A2x2 − x2 + A−2 + A−4

S2,∞(561) = A2x2 − 4A−2x2 + A−2 − A−4x2 + A−4 + 4A−6x2 − A−6 + 2A−8x2 − A−8 − 2A−10x2 − A−10 −A−12x2 − A−12 + A−14x2 + A−16 − A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 − A8x2 + A8 + 4A6x2 − A6 + 2A4x2 − A4 − 2A2x2 − A2 −x2 − 1 + A−2 + A−4

S2,∞(562) = A8x − 2A4x3 + 5x3 − 4x − 3A−4x3 + A−4x + A−8x3 − A−12xS2,∞(562) = −A12x + 2A8x3 − A8x − 4A4x3 + 2A4x + 4x3 − 3x − A−4x3 − A−4x + A−8xS2,∞(563) = −A6x2 + 3A2x2 −A2 + x2 − 1−4A−2x2 +A−2 −2A−4x2 +A−4 +2A−6x2 +A−6 +A−8x2 +A−8 −A−10x2 − A−12 + A−14

S2,∞(563) = A14 − 2A10x2 +A10 +4A6x2 −A6 +A4x2 −A4 −4A2x2 +A2 − 2x2 + 1+A−2x2 +A−2 +A−4x2 +A−4 − A−6 − A−8

S2,∞(564) = A8x − 2A4x3 + A4x + 4x3 − 2x − 3A−4x3 + 2A−4x + A−8x3 − A−12xS2,∞(564) = −A12x + 2A8x3 − A8x − 4A4x3 + 3A4x + 3x3 − x − A−4x3 + A−8xS2,∞(565) = −A6x2 + 5A2x2 − 2A2 + 2x2 − 2− 6A−2x2 +A−2 − 4A−4x2 + 2A−4 + 3A−6x2 +A−6 + 2A−8x2 +A−8 − A−10x2 − A−10 − 3A−12 + A−14

S2,∞(565) = A14 − 3A10x2 +A10 + 7A6x2 − 3A6 + 3A4x2 − 3A4 − 5A2x2 − 4x2 + 1+A−2x2 + 2A−2 +A−4x2 +2A−4 − A−6 − 2A−8

S2,∞(566) = A10x2 − 2A6x2 + A6 − A4x2 + A4 + 2A2x2 + x2 − 2A−2x2 − A−4x2 + A−6x2 + A−8 − A−10


S2,∞(566) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − x2 + 1 + 2A−2x2 + A−4x2 − A−6x2 − A−8x2 + A−10 + A−12

S2,∞(567) = −A−6x2 + A−6 − A−8x2 + A−8 + A−10x2 + A−12x2 − A−14 − A−16x2 + A−20

S2,∞(567) = A18x2 − A18 − A14x2 − A12x2 + A12 + A10x2 + A8x2 − A4x2 − A2x2 + A2 + 1S2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x − A−4xS2,∞(568) = A4x3 − 2A4x − x3 + x + A−4x3 − 2A−4xS2,∞(569) = A−2x2 − A−2 + A−4x2 − A−4 − A−6x2 − 2A−8x2 + A−8 + A−10 + A−12x2 − 2A−16

S2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 + 2A8x2 − 2A8 − A6 − 2A4x2 − A2x2 + A2 + 1 − A−4

S2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x + A4x3 − A4x + xS2,∞(570) = −x3 + 2x + 2A−4x3 − 2A−4x − A−8x3 + A−8xS2,∞(571) = A8x2 − A8 − A4x2 − A2x2 + 1 + A−2x2 − A−2 − A−6

S2,∞(571) = −A6+A2x2−A2−A−2−A−6x2+A−6−A−8x2+A−8−A−10x2+A−10+A−12x2+A−14x2−A−14−A−16

S2,∞(572) = A−8x3 − A−8x − A−12x3 + 2A−12xS2,∞(572) = A16x3 − A16x − 2A12x3 + 3A12x + A4x3 − A4xS2,∞(573) = A6x2 + A4x2 − A4 − A2x2 − x2 − A−8

S2,∞(573) = A2x2 − A2 + x2 − 1 − A−2x2 − A−4x2 + A−6 − A−12

S2,∞(574) = −A22 + A18x2 − A8x2 + A8 − A6x2 + A6 + A4

S2,∞(575) = A2 − A−6x2 + A−6 − A−8x2 + A−8 − A−10x2 + A−10 + A−12x2 + A−14x2 − A−14 − A−16

S2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 − A−4x2 + A−4 + A−6 + A−8 − A−10

S2,∞(577) = −A10 + A6x2 − A2x2 − x2 + 1 + A−2x2 + A−4 − A−6x2 + A−6

S2,∞(579) = −A−6x2+A−6+A−10x2−2A−14x2+A−14−A−16x2+A−16+A−18x2+A−18+A−20x2−A−22−A−24

S2,∞(579) = −A22x2 + 2A18x2 + A16x2 − A16 − A14x2 − A12x2 + A10 + A8 − A6x2 + A6

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 − x2 + 1 + A−2 + A−4

S2,∞(580) = A2x2 − A−2x2 − A−4x2 + A−4 + A−6 + A−8 − A−10x2 + A−14x2 − A−14

S2,∞(581) = −A−12x2 + A−12 − A−14x2 + A−16 + 2A−18x2 − A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A8x2 + A8 − A6x2 + A6 + A4

S2,∞(582) = −A−6x2 − A−8x2 + A−8 + 2A−10x2 − A−10 + A−12x2 − A−14x2 − A−16x2 + A−18 + A−20

S2,∞(582) = A18x2 − 2A14x2 + A14 − A12x2 + A12 + 2A10x2 − A10 + A8x2 − A6 − A4x2 − A2x2 + A2 + 1S2,∞(583) = −A4x3 + A4x + 2x3 − x − A−4x3 + A−4xS2,∞(583) = A8x3 − A8x − 2A4x3 + 2A4x + x3

S2,∞(584) = A−2x2 + A−4x2 − A−4 − 2A−6x2 + A−6 − 2A−8x2 + A−8 + A−10x2 + A−12x2 − A−14 − 2A−16

S2,∞(584) = −A14x2 + 3A10x2 − 2A10 + 2A8x2 − 2A8 − A6x2 − 2A4x2 − A2x2 + 2A2 + 1 − A−4

S2,∞(585) = A12x2 − A12 + A10x2 − A10 − A8 − A4x2 − A2x2 + A2

Knots in L(7, 1)

S2,∞(32) = A6x2 − A2x2 − Ax3 + 2Ax + A−3xS2,∞(32) = A2x2 − A2 − A−2x2 − A−3x3 + 2A−3x + A−6 + A−7xS2,∞(35) = A2x2 − 2A−2x2 + A−2 − A−3x3 + 2A−3x + A−6x2 + A−7x − A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 − Ax3 + 2Ax + A−2 + A−3xS2,∞(36) = A10x2 − A10 − A5x3 + 2A5x − A2x2 + A2 + AxS2,∞(41) = −A−6x2 + A−6 − A−7x3 + 2A−7x + A−10 + A−11x3 − A−11x − A−15xS2,∞(41) = −A14x2 + A14 + A10x2 + A9x3 − 2A9x − A5x3 + A5x − A2x2 + A2 + AxS2,∞(42) = −A8x3 + A8x + A4x3 − A4x + Ax2 − A− A−3


S2,∞(42) = −x3 + 2x + A−4x3 − A−4x + A−7x2 − A−7 − A−8x − A−11

S2,∞(44) = A2x2 − 3A−2x2 + 2A−2 − A−3x3 + 2A−3x + A−6x2 + A−6 + A−7x3 − A−7x − A−10 − A−11xS2,∞(44) = −A10x2 + 2A6x2 + A5x3 − 2A5x − 2A2x2 + A2 − Ax3 + Ax + A−2 + A−3xS2,∞(45) = −A4x3 + 2x3 − x + A−3x2 − A−3 − A−4x3 − A−7 + A−8xS2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x + A−3x2 − A−3 − 2A−4x − A−7

S2,∞(412) = A2x2 − 3A−2x2 + A−2 − A−3x3 + 2A−3x + 2A−6x2 + A−7x3 − A−7x − A−10x2 − A−11x + A−14

S2,∞(412) = A14 − 2A10x2 + A10 + 3A6x2 − A6 + A5x3 − 2A5x − 2A2x2 − Ax3 + Ax + A−2 + A−3xS2,∞(414) = A8x − 2A4x3 + A4x + 3x3 − 2x + A−3x2 − A−3 − A−4x3 − A−4x − A−7 + A−8xS2,∞(415) = −A6x2 + 2A2x2 − A2 + Ax3 − 2Ax − 2A−2x2 − A−3x3 + A−3x + A−6x2 + A−7x − A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − Ax3 + 2Ax + A−2x2 + A−3x3 − A−3x − A−6 − A−7xS2,∞(417) = A6x2 − 2A2x2 + A2 − Ax3 + 2Ax + A−2 + A−3x3 − A−3x − A−7xS2,∞(417) = −A6x2 + A6 + A2x2 + Ax3 − 2Ax − A−2x2 − A−3x3 + A−3x + A−6 + A−7xS2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x + A−7x2 − A−7 − A−8x3 − A−11 + A−12xS2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x + Ax2 − A− x − A−3

S2,∞(419) = −A10x2 + 2A6x2 − A6 + A5x3 − 2A5x − A2x2 − A2 − Ax3 + Ax + A−3xS2,∞(419) = A2x2 − A2 − 2A−2x2 − A−3x3 + 2A−3x + A−6x2 + A−7x3 − A−7x − A−10 − A−11xS2,∞(420) = −A−6x2 − A−7x3 + 2A−7x + A−10x2 − A−10 + A−11x3 − A−11x − A−14 − A−15xS2,∞(420) = −A14x2 + 2A10x2 − 2A10 + A9x3 − 2A9x − A6 − A5x3 + A5x − A2x2 + A2 + AxS2,∞(421) = −A4x3 + A4x + x3 + A−3x2 − A−3 − A−4x − A−7

S2,∞(424) = −A12x3 + 2A12x + A5x2 − A5 + A4x3 − 2A4x − AS2,∞(52) = A−6 − 2A−10x2 + A−10 − A−11x3 + 2A−11x + A−14x2 + A−15x3 − A−15x − A−19xS2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 + A13x3 − 2A13x − A10x2 − A9x3 + A9x − A6x2 + 2A6 + A5xS2,∞(54) = A14x2 − A14 − A6x2 + A6 − A5x3 + 2A5x + AxS2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 − A−7x3 + 2A−7x − A−10x2 + A−10 + A−11x + A−14x2 − A−14

S2,∞(55) = A−8x3 − 2A−8x + A−11x2 − A−11 − A−12x − A−15x2 + A−19

S2,∞(55) = A16x3 − 2A16x − A12x3 + A12x − A9x2 + A9 + A5x2 + A4x3 − 2A4x − AS2,∞(56) = A10x2 − A10 + A9x3 − 2A9x − A5x3 + A5x − A2x2 + A2

S2,∞(57) = −2A−6x2+A−6−A−7x3+2A−7x +2A−10x2−A−10+2A−11x3−3A−11x −2A−14−A−15x3+A−19xS2,∞(57) = A18x2−A18−2A14x2−A13x3+2A13x+2A10x2−A10+2A9x3−3A9x−A6−A5x3−A2x2+A2+AxS2,∞(58) = −x3 + x + 3A−4x3 − 4A−4x + A−7x2 − A−7 − A−8x3 − A−8x − A−11x2 + A−12x + A−15

S2,∞(58) = A12x3 − A12x − 2A8x3 + A8x − A5x2 + A5 + 2A4x3 − 2A4x + Ax2 − x − A−3

S2,∞(59) = 2A6x2 − A6 + A5x3 − 2A5x − 2A2x2 − 2Ax3 + 3Ax + A−2 + A−3x3 − A−3x − A−7xS2,∞(59) = −A6x2 + A6 + 2A2x2 − A2 + 2Ax3 − 4Ax − A−2x2 − A−2 − 2A−3x3 + 2A−3x + A−6 + A−7xS2,∞(511) = A2x2−4A−2x2+2A−2−A−3x3+2A−3x+3A−6x2+2A−7x3−3A−7x−A−10x2−A−10−A−11x3+A−14 + A−15xS2,∞(511) = A14x2 − 3A10x2 + A10 − A9x3 + 2A9x + 3A6x2 + 2A5x3 − 3A5x − 2A2x2 − Ax3 + A−2 + A−3xS2,∞(512) = −x3+4A−4x3−4A−4x+A−7x2−A−7−3A−8x3+A−8x−A−11x2+A−12x3+A−12x+A−15−A−16xS2,∞(512) = −A16x + 2A12x3 − A12x − 4A8x3 + 3A8x − A5x2 + A5 + 3A4x3 − 2A4x + Ax2 − 2x − A−3

S2,∞(513) = −A6x2+3A2x2−2A2+Ax3−2Ax−3A−2x2−2A−3x3+3A−3x+A−6x2+A−6+A−7x3−A−10−A−11xS2,∞(513) = −A10x2 + 3A6x2 − A6 + A5x3 − 2A5x − 3A2x2 − 2Ax3 + 3Ax + A−2x2 + A−3x3 − A−6 − A−7xS2,∞(514) = −A4x3 + 3x3 − 2x + A−3x2 − A−3 − 2A−4x3 − A−7x2 + A−8x3 + A−11 − A−12xS2,∞(514) = −A12x + 2A8x3 − 2A8x − 3A4x3 + 2A4x − Ax2 + A+ 2x3 + A−3x2 − 2A−4x − A−7

S2,∞(515) = −A10x2+4A6x2−A6+2A5x3−4A5x−4A2x2+A2−3Ax3+4Ax+A−2x2+A−2+A−3x3−A−6−A−7xS2,∞(515) = −A6x2 + 4A2x2 − 2A2 + 2Ax3 − 4Ax − 4A−2x2 + A−2 − 3A−3x3 + 4A−3x + A−6x2 + 2A−6 +A−7x3 − A−10 − A−11x


S2,∞(516) = A8x3 − 3A4x3 + 2A4x − Ax2 + A+ 3x3 − x + A−3x2 − A−4x3 − A−4x − A−7 + A−8xS2,∞(516) = A8x − 2A4x3 + A4x + 4x3 − 3x + A−3x2 − A−3 − 2A−4x3 − A−7x2 + 2A−8x + A−11

S2,∞(517) = A14x2 − 2A10x2 −A9x3 + 2A9x + 3A6x2 −A6 +A5x3 −A5x − 2A2x2 −Ax3 +Ax +A−2 +A−3xS2,∞(517) = A2x2 − 3A−2x2 + A−2 − A−3x3 + 2A−3x + 3A−6x2 − A−6 + A−7x3 − A−7x − A−10x2 − A−10 −A−11x3 + A−11x + A−14 + A−15xS2,∞(518) = −2A−6x2+A−6−A−7x3+2A−7x+2A−10x2+2A−11x3−3A−11x−A−14x2−A−15x3+A−18+A−19xS2,∞(518) = A18x2−3A14x2+2A14−A13x3+2A13x +2A10x2+2A9x3−3A9x −A6−A5x3−A2x2+A2+AxS2,∞(519) = −A4x3 + 3x3 − 3x + A−3x2 − A−3 − A−4x3 − A−4x − A−7x2 + A−8x + A−11

S2,∞(519) = A8x3 − A8x − 2A4x3 + A4x − Ax2 + A+ 2x3 − x + A−3x2 − 2A−4x − A−7

S2,∞(520) = 2A2x2 −A2 +Ax3 − 2Ax − 3A−2x2 +A−2 − 2A−3x3 + 3A−3x +A−6x2 +A−6 +A−7x3 −A−7x −A−10 − A−11xS2,∞(520) = −A10x2 + 3A6x2 − A6 + 2A5x3 − 4A5x − 2A2x2 − 2Ax3 + 2Ax + A−2 + A−3xS2,∞(521) = A12x3 − 3A8x3 + 3A8x − A5x2 + A5 + 2A4x3 − A4x + Ax2 − x − A−3

S2,∞(521) = −x3 + x + 3A−4x3 − 3A−4x + A−7x2 − A−7 − 2A−8x3 + A−8x − A−11x2 + 2A−12x + A−15

S2,∞(522) = −A6x2 + 2A2x2 − A2 + Ax3 − 2Ax − 2A−2x2 + A−2 − A−3x3 + A−3x + 2A−6 + A−7xS2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 − Ax3 + 2Ax + A−2x2 + A−3x3 − A−3x − A−6 − A−7xS2,∞(524) = A−6 − A−7x3 + 2A−7x − A−10x2 + A−10 + A−11x + A−14x2 − A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 + A6 − A5x3 + 2A5x − A2x2 + A2 + AxS2,∞(525) = −A−6x2+2A−10x2−4A−14x2+2A−14−A−15x3+2A−15x+3A−18x2+A−19x3−A−19x−A−22x2−A−22 − A−23x + A−26

S2,∞(525) = A26−2A22x2+4A18x2−A18+A17x3−2A17x−3A14x2+A14−A13x3+A13x+A10x2+A10+A9x−A6x2

S2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 − Ax3 + 2Ax + A−2x2 + A−2 + A−3x − A−6

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 − A−3x3 + 2A−3x + 2A−6x2 + A−7x − 2A−10x2 + A−14x2

S2,∞(534) = −A6x2+3A2x2−A2−3A−2x2+A−2−A−3x3+2A−3x +2A−6x2+A−7x −A−10x2−A−10+A−14

S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 − Ax3 + 2Ax + 2A−2x2 + A−3x − A−6x2

S2,∞(535) = −A−6x2 + 2A−10x2 − A−10 − 3A−14x2 − A−15x3 + 2A−15x + 3A−18x2 − A−18 + A−19x3 − A−19x −A−22x2 − A−22 − A−23x + A−26

S2,∞(535) = A26−2A22x2+4A18x2−2A18+A17x3−2A17x−2A14x2−A14−A13x3+A13x+A10x2+A9x−A6x2

S2,∞(538) = −A−2 + A−6x2 − A−6 − 2A−10x2 − A−11x3 + 2A−11x + A−14x2 + A−15x3 − A−15x − A−19xS2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 + A13x3 − 2A13x − A10x2 − A10 − A9x3 + A9x + A5x − A2

S2,∞(541) = A18 + A10x2 − A10 − A6x2 − A5x3 + 2A5x + AxS2,∞(541) = A−2x2 − A−2 − A−6x2 − A−7x3 + 2A−7x + A−11x + A−18

S2,∞(543) = A6x2 − 2A2x2 + A2 − Ax3 + 2Ax + A−2x2 + A−3x3 − A−3x − A−6 − A−7x3 + A−7x + A−11xS2,∞(543) = A10x2 −A10 −A6x2 −A5x3 + 2A5x +A2x2 +Ax3 −Ax −A−2x2 −A−3x3 +A−3x +A−6 +A−7xS2,∞(544) = −x3 + x + 3A−4x3 − 3A−4x + A−7x2 − A−7 − 2A−8x3 − A−11x2 + A−12x3 + A−15 − A−16xS2,∞(544) = −A16x + 2A12x3 − 2A12x − 3A8x3 + 2A8x − A5x2 + A5 + 2A4x3 − A4x + Ax2 − x − A−3

S2,∞(545) = −A10x2 + 3A6x2 − A6 + A5x3 − 2A5x − 3A2x2 + A2 − 2Ax3 + 3Ax + 2A−2 + A−3x3 − A−7xS2,∞(545) = −A6x2 +A6 +2A2x2 +Ax3 −2Ax − 3A−2x2 +A−2 −2A−3x3 + 3A−3x +A−6x2 +A−6 +A−7x3 −A−10 − A−11xS2,∞(546) = −x3+x+3A−4x3−2A−4x+A−7x2−A−7−3A−8x3+2A−8x−A−11x2+A−12x3+A−12x+A−15−A−16xS2,∞(546) = −A16x + 2A12x3 − A12x − 4A8x3 + 4A8x − A5x2 + A5 + 2A4x3 + Ax2 − x − A−3

S2,∞(547) = −A10x2 + 4A6x2 − 2A6 + 2A5x3 − 4A5x − 3A2x2 − 3Ax3 + 4Ax + 2A−2 + A−3x3 − A−7xS2,∞(547) = −A6x2+A6+3A2x2−A2+2Ax3−4Ax −3A−2x2−3A−3x3+4A−3x +A−6x2+A−6+A−7x3−A−10 − A−11xS2,∞(548) = A4x3 − A4x − 2x3 + 2x − A−3x2 + A−3 + 2A−4x3 − A−4x + A−7x2 − A−8x3 − A−11 + A−12x


S2,∞(548) = A12x − 2A8x3 + 2A8x + 3A4x3 − 3A4x + Ax2 − A− x3 − A−3x2 + A−4x + A−7

S2,∞(549) = A14x2 − 2A10x2 + A10 − A9x3 + 2A9x + 2A6x2 + A5x3 − A5x − A2x2 − A2 − Ax3 + Ax + A−3xS2,∞(549) = A2x2 −A2 − 2A−2x2 −A−3x3 + 2A−3x + 2A−6x2 +A−7x3 −A−7x −A−10x2 −A−11x3 +A−11x +A−14 + A−15xS2,∞(553) = A−2x2 − 2A−6x2 + A−6 − A−7x3 + 2A−7x + A−10x2 + A−11x − A−14x2 + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 − A5x3 + 2A5x + A2 + AxS2,∞(555) = −A18x2+3A14x2−A14+A13x3−2A13x−3A10x2+A10−A9x3+A9x+A6x2+A6+A5x−A2x2+A−2

S2,∞(555) = A2−A−2x2+2A−6x2−4A−10x2+2A−10−A−11x3+2A−11x+2A−14x2+A−15x3−A−15x−A−18−A−19xS2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 − Ax3 + 2Ax + A−2x2 + A−3x − A−6

S2,∞(556) = −A6+2A2x2−A2−2A−2x2−A−3x3+2A−3x+A−6x2+A−6+A−7x−2A−10x2+A−10+A−14x2

S2,∞(559) = A2x2−3A−2x2+A−2−A−3x3+2A−3x +3A−6x2+A−7x3−A−7x −2A−10x2−A−11x3+A−11x +A−14x2 + A−15x − A−18

S2,∞(559) = −A18 + 2A14x2 − A14 − 3A10x2 + A10 − A9x3 + 2A9x + 3A6x2 + A5x3 − A5x − 2A2x2 − Ax3 +Ax + A−2 + A−3xS2,∞(561) = A2x2 − 4A−2x2 +A−2 −A−3x3 + 2A−3x + 4A−6x2 −A−6 + 2A−7x3 − 3A−7x − 2A−10x2 −A−10 −A−11x3 + A−14x2 + A−15x − A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 − A9x3 + 2A9x + 4A6x2 − A6 + 2A5x3 − 3A5x − 2A2x2 −A2 − Ax3 + A−2 + A−3xS2,∞(562) = A8x − 2A4x3 + 5x3 − 4x + A−3x2 − A−3 − 3A−4x3 − A−7x2 + A−8x3 + A−8x + A−11 − A−12xS2,∞(562) = −A12x+2A8x3−A8x−4A4x3+2A4x−Ax2+A+4x3−2x+A−3x2−A−4x3−2A−4x−A−7+A−8xS2,∞(563) = −A6x2+3A2x2−A2+Ax3−2Ax−4A−2x2+A−2−2A−3x3+3A−3x+2A−6x2+A−6+A−7x3−A−10x2 − A−11x + A−14

S2,∞(563) = A14 − 2A10x2 +A10 + 4A6x2 −A6 +A5x3 − 2A5x − 4A2x2 +A2 − 2Ax3 + 3Ax +A−2x2 +A−2 +A−3x3 − A−6 − A−7xS2,∞(564) = A8x−2A4x3+A4x+4x3−2x+A−3x2−A−3−3A−4x3+A−4x−A−7x2+A−8x3+A−8x+A−11−A−12xS2,∞(564) = −A12x + 2A8x3 −A8x −4A4x3 + 3A4x −Ax2 +A+ 3x3 +A−3x2 −A−4x3 −A−4x −A−7 +A−8xS2,∞(565) = −A6x2 + 5A2x2 − 2A2 + 2Ax3 − 4Ax − 6A−2x2 + A−2 − 4A−3x3 + 6A−3x + 3A−6x2 + A−6 +2A−7x3 − A−7x − A−10x2 − A−10 − 2A−11x + A−14

S2,∞(565) = A14 − 3A10x2 + A10 + 7A6x2 − 3A6 + 3A5x3 − 6A5x − 5A2x2 − 4Ax3 + 5Ax + A−2x2 + 2A−2 +A−3x3 + A−3x − A−6 − A−7xS2,∞(566) = A10x2 − 2A6x2 +A6 −A5x3 + 2A5x + 2A2x2 +Ax3 −Ax − 2A−2x2 −A−3x3 +A−3x +A−6x2 +A−7x − A−10

S2,∞(566) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 − Ax3 + 2Ax + 2A−2x2 + A−3x3 − A−3x − A−6x2 − A−7x3 +A−7x + A−10 + A−11xS2,∞(567) = −A−6x2 + A−6 − A−7x3 + 2A−7x + A−10x2 + A−11x3 − A−11x − A−14 − A−15x3 + A−15x + A−19xS2,∞(567) = A18x2 − A18 − A14x2 − A13x3 + 2A13x + A10x2 + A9x3 − A9x − A5x3 + A5x − A2x2 + A2 + AxS2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x + Ax2 − A− x − A−3x2 + A−7

S2,∞(568) = A4x3 − 2A4x − x3 + x − A−3x2 + A−3 + A−4x3 − A−4x + A−7x2 − A−8x − A−11

S2,∞(569) = A−2x2 − A−2 + A−3x3 − 2A−3x − A−6x2 − 2A−7x3 + 3A−7x + A−10 + A−11x3 − A−11x − A−15xS2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 + 2A9x3 − 4A9x − A6 − 2A5x3 + 2A5x − A2x2 + A2 + AxS2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x − A5x2 + A5 + A4x3 + Ax2 − A−3

S2,∞(570) = −x3 + 2x + 2A−4x3 − 2A−4x + A−7x2 − A−7 − A−8x3 − A−11x2 + A−12x + A−15

S2,∞(571) = A9x3 − 2A9x − A5x3 + A5x − A2x2 + Ax + A−2x2 − A−2 − A−6

S2,∞(571) = −A6 + A2x2 − A2 − A−2 − A−6x2 + A−6 − A−7x3 + 2A−7x − A−10x2 + A−10 + A−11x3 − A−11x +A−14x2 − A−14 − A−15x


S2,∞(572) = A−8x3 − A−8x + A−11x2 − A−11 − A−12x3 + A−12x − A−15x2 + A−16x + A−19

S2,∞(572) = A16x3 − A16x − 2A12x3 + 3A12x − A9x2 + A9 + A8x + A5x2 + A4x3 − 2A4x − AS2,∞(573) = A6x2 + A5x3 − 2A5x − A2x2 − Ax3 + AxS2,∞(573) = A2x2 − A2 + Ax3 − 2Ax − A−2x2 − A−3x3 + A−3x + A−6

S2,∞(574) = −A22 + A18x2 − A9x3 + 2A9x − A6x2 + A6 + A5xS2,∞(575) = A2 −A−6x2 +A−6 −A−7x3 + 2A−7x −A−10x2 +A−10 +A−11x3 −A−11x +A−14x2 −A−14 −A−15xS2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 − A−3x3 + 2A−3x + A−6 + A−7x − A−10

S2,∞(577) = −A10 + A6x2 − A2x2 − Ax3 + 2Ax + A−2x2 + A−3x − A−6x2 + A−6

S2,∞(579) = −A−6x2+A−6+A−10x2−2A−14x2+A−14−A−15x3+2A−15x+A−18x2+A−18+A−19x3−A−19x−A−22 − A−23xS2,∞(579) = −A22x2 + 2A18x2 + A17x3 − 2A17x − A14x2 − A13x3 + A13x + A10 + A9x − A6x2 + A6

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 − Ax3 + 2Ax + A−2 + A−3xS2,∞(580) = A2x2 − A−2x2 − A−3x3 + 2A−3x + A−6 + A−7x − A−10x2 + A−14x2 − A−14

S2,∞(581) = −A−11x3 + 2A−11x − A−14x2 + A−15x + 2A−18x2 − A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 − A9x3 + 2A9x − A6x2 + A6 + A5xS2,∞(582) = −A−6x2 − A−7x3 + 2A−7x + 2A−10x2 − A−10 + A−11x3 − A−11x − A−14x2 − A−15x3 + A−15x +A−18 + A−19xS2,∞(582) = A18x2−2A14x2+A14−A13x3+2A13x+2A10x2−A10+A9x3−A9x−A6−A5x3+A5x−A2x2+A2+AxS2,∞(583) = −A4x3 + A4x + 2x3 − x + A−3x2 − A−3 − A−4x3 − A−7x2 + A−8x + A−11

S2,∞(583) = A8x3 − A8x − 2A4x3 + 2A4x − Ax2 + A+ x3 + x + A−3x2 − A−4x − A−7

S2,∞(584) = A−2x2+A−3x3−2A−3x−2A−6x2+A−6−2A−7x3+3A−7x+A−10x2+A−11x3−A−11x−A−14−A−15xS2,∞(584) = −A14x2 + 3A10x2 − 2A10 + 2A9x3 − 4A9x − A6x2 − 2A5x3 + 2A5x − A2x2 + 2A2 + AxS2,∞(585) = −A−15xS2,∞(585) = A13x3 − 2A13x + A10x2 − A10 − A9x − A5x3 + A5x − A2x2 + A2 + Ax

Knots in L(7, 2)

S2,∞(32) = A6x2 − A2x2 + A−2x3 − 2A−2x − A−6xS2,∞(32) = A2x2 − A2 − A−2x2 + A−6x3 − 2A−6x + A−6 − A−10xS2,∞(35) = A2x2 − 2A−2x2 + A−2 + A−6x3 + A−6x2 − 2A−6x − A−10x − A−10

S2,∞(35) = −A10 + 2A6x2 − A6 − 2A2x2 + A2 + A−2x3 − 2A−2x + A−2 − A−6xS2,∞(36) = A10x2 − A10 + A2x3 − A2x2 − 2A2x + A2 − A−2xS2,∞(41) = −A−6x2 + A−6 + A−10x3 − 2A−10x + A−10 − A−14x3 + A−14x + A−18xS2,∞(41) = −A14x2 + A14 + A10x2 − A6x3 + 2A6x + A2x3 − A2x2 − A2x + A2 − A−2xS2,∞(42) = −A8x3 + A8x + A4x3 − A4x − x3 + 2x + A−4x3 − A−4x2 − A−4x + A−4 − A−8x + A−8

S2,∞(42) = −x3 + 2x + A−4x3 − A−4x − A−8x3 + A−8x + A−12x3 − A−12x2 − A−12x + A−12 − A−16x + A−16

S2,∞(44) = A2x2 − 3A−2x2 + 2A−2 + A−6x3 + A−6x2 − 2A−6x + A−6 − A−10x3 + A−10x − A−10 + A−14xS2,∞(44) = −A10x2 + 2A6x2 − A2x3 − 2A2x2 + 2A2x + A2 + A−2x3 − A−2x + A−2 − A−6xS2,∞(45) = −A4x3 + 2x3 − x − 2A−4x3 + 2A−4x + A−8x3 − A−8x2 + A−8 − A−12x + A−12

S2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x − A−4x3 + A−8x3 − A−8x2 − A−8x + A−8 − A−12x + A−12

S2,∞(412) = A2x2 − 3A−2x2 + A−2 + A−6x3 + 2A−6x2 − 2A−6x − A−10x3 − A−10x2 + A−10x + A−14x + A−14

S2,∞(412) = A14 − 2A10x2 + A10 + 3A6x2 − A6 − A2x3 − 2A2x2 + 2A2x + A−2x3 − A−2x + A−2 − A−6xS2,∞(414) = A8x − 2A4x3 + A4x + 3x3 − 2x − 2A−4x3 + A−4x + A−8x3 − A−8x2 + A−8 − A−12x + A−12


S2,∞(415) = −A6x2 + 2A2x2 − A2 − A−2x3 − 2A−2x2 + 2A−2x + A−6x3 + A−6x2 − A−6x − A−10x − A−10

S2,∞(415) = −A10 + 2A6x2 − A6 − 3A2x2 + A2 + A−2x3 + A−2x2 − 2A−2x − A−6x3 + A−6x − A−6 + A−10xS2,∞(417) = A6x2 − 2A2x2 + A2 + A−2x3 − 2A−2x + A−2 − A−6x3 + A−6x + A−10xS2,∞(417) = −A6x2 + A6 + A2x2 − A−2x3 − A−2x2 + 2A−2x + A−6x3 − A−6x + A−6 − A−10xS2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x − 2A−8x3 + 2A−8x + A−12x3 − A−12x2 + A−12 − A−16x + A−16

S2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x − x3 + x + A−4x3 − A−4x2 − A−4x + A−4 − A−8x + A−8

S2,∞(419) = −A10x2 + 2A6x2 − A6 − A2x3 − A2x2 + 2A2x − A2 + A−2x3 − A−2x − A−6xS2,∞(419) = A2x2 − A2 − 2A−2x2 + A−6x3 + A−6x2 − 2A−6x − A−10x3 + A−10x − A−10 + A−14xS2,∞(420) = −A−6x2 + A−10x3 + A−10x2 − 2A−10x − A−10 − A−14x3 + A−14x − A−14 + A−18xS2,∞(420) = −A14x2 + 2A10x2 − 2A10 − A6x3 + 2A6x − A6 + A2x3 − A2x2 − A2x + A2 − A−2xS2,∞(421) = −A4x3 + A4x + x3 − A−4x3 + A−4x + A−8x3 − A−8x2 − A−8x + A−8 − A−12x + A−12

S2,∞(424) = −A−12x3 + 2A−12x + A−16x3 − A−16x2 − A−16x + A−16 − A−20x + A−20

S2,∞(424) = −A12x3 + 2A12x + x3 − x2 − x + 1 − A−4x + A−4

S2,∞(52) = A−6 − 2A−10x2 + A−10 + A−14x3 + A−14x2 − 2A−14x − A−18x3 + A−18x + A−22xS2,∞(52) = −A18x2 + A18 + 2A14x2 − A14 − A10x3 − A10x2 + 2A10x + A6x3 − A6x2 − A6x + 2A6 − A2xS2,∞(54) = A14x2 − A14 − A6x2 + A6 + A2x3 − 2A2x − A−2xS2,∞(54) = A−2x2 − A−2 − A−6x2 + A−6 + A−10x3 − A−10x2 − 2A−10x + A−10 + A−14x2 − A−14x − A−14

S2,∞(55) = A−8x3 − 2A−8x − A−12x3 + A−12x + 2A−16x3 − A−16x2 − 3A−16x + A−16 − A−20x3 + A−20x2 +A−24x − A−24

S2,∞(55) = A16x3 − 2A16x −A12x3 +A12x +A8x3 − 2A8x −A4x3 +A4x2 +A4x −A4 + x3 − x2 −A−4x +A−4

S2,∞(56) = A10x2−A10−A6x3+2A6x +A2x3−A2x2−A2x +A2−A−2x3+A−2x2+A−2x −A−2+A−6x3−A−6x2 − A−10x + A−10

S2,∞(56) = −A−6x3 + 2A−6x + A−10x3 − A−10x − A−14x3 + A−14x2 + A−14x − A−14 + A−18x3 − A−18x2 −A−22x + A−22

S2,∞(57) = −2A−6x2+A−6+A−10x3+2A−10x2−2A−10x−A−10−2A−14x3+3A−14x−2A−14+A−18x3−A−22xS2,∞(57) = A18x2−A18−2A14x2+A10x3+2A10x2−2A10x−A10−2A6x3+3A6x−A6+A2x3−A2x2+A2−A−2xS2,∞(58) = −x3 + x + 3A−4x3 − 4A−4x − 2A−8x3 + A−8x + 2A−12x3 − A−12x2 − 2A−12x + A−12 − A−16x3 +A−16x2 + A−20x − A−20

S2,∞(58) = A12x3 −A12x −2A8x3 +A8x + 3A4x3 −4A4x −2x3 + x2 +2x − 1+A−4x3 −A−4x2 −A−8x +A−8

S2,∞(59) = 2A6x2 − A6 − A2x3 − 2A2x2 + 2A2x + 2A−2x3 − 3A−2x + A−2 − 2A−6x3 + A−6x2 + 2A−6x −A−6 + A−10x3 − A−10x2 + A−10x − A−14x + A−14

S2,∞(59) = −A6x2+A6+2A2x2−A2−2A−2x3−A−2x2+4A−2x −A−2+2A−6x3−2A−6x +A−6−A−10x3+A−10x2 − A−10 + A−14x3 − A−14x2 − A−18x + A−18

S2,∞(511) = A2x2 − 4A−2x2 + 2A−2 + A−6x3 + 3A−6x2 − 2A−6x − 2A−10x3 − A−10x2 + 3A−10x − A−10 +A−14x3 + A−14 − A−18xS2,∞(511) = A14x2 − 3A10x2 + A10 + A6x3 + 3A6x2 − 2A6x − 2A2x3 − 2A2x2 + 3A2x + A−2x3 + A−2 − A−6xS2,∞(512) = −x3 + 4A−4x3 − 4A−4x − 4A−8x3 + 3A−8x + 3A−12x3 − A−12x2 − 2A−12x + A−12 − A−16x3 +A−16x2 − A−16x + A−20x − A−20

S2,∞(512) = −A16x + 2A12x3 −A12x − 4A8x3 + 3A8x + 4A4x3 − 4A4x − 2x3 + x2 + x − 1+A−4x3 −A−4x2 −A−8x + A−8

S2,∞(513) = −A6x2 + 3A2x2 −2A2 −A−2x3 − 3A−2x2 +2A−2x +2A−6x3 +A−6x2 − 3A−6x +A−6 −A−10x3 −A−10 + A−14xS2,∞(513) = −A10x2+3A6x2−A6−A2x3−3A2x2+2A2x+2A−2x3+A−2x2−3A−2x−A−6x3−A−6+A−10xS2,∞(514) = −A4x3 + 3x3 − 2x − 3A−4x3 + 2A−4x + 3A−8x3 − A−8x2 − 3A−8x + A−8 − A−12x3 + A−12x2 −A−12x + A−16x − A−16


S2,∞(514) = −A12x + 2A8x3 − 2A8x − 3A4x3 + 2A4x + 3x3 − 2x − 2A−4x3 +A−4x2 +A−4x −A−4 +A−8x3 −A−8x2 − A−12x + A−12

S2,∞(515) = −A10x2+4A6x2−A6−2A2x3−4A2x2+4A2x+A2+3A−2x3+A−2x2−4A−2x+A−2−2A−6x3+A−6x2 + A−6x − 2A−6 + A−10x3 − A−10x2 + A−10x − A−14x + A−14

S2,∞(515) = −A6x2 +4A2x2 − 2A2 − 2A−2x3 −4A−2x2 +4A−2x +A−2 + 3A−6x3 +A−6x2 −4A−6x + 2A−6 −2A−10x3 + A−10x2 + A−10x − 2A−10 + A−14x3 − A−14x2 + A−14x − A−18x + A−18

S2,∞(516) = A8x3 − 3A4x3 + 2A4x + 4x3 − 3x − 3A−4x3 +A−4x2 + 2A−4x −A−4 +A−8x3 −A−8x2 +A−8x −A−12x + A−12

S2,∞(516) = A8x − 2A4x3 +A4x + 4x3 − 3x − 3A−4x3 + 2A−4x + 2A−8x3 −A−8x2 −A−8x +A−8 −A−12x3 +A−12x2 + A−16x − A−16

S2,∞(517) = A14x2−2A10x2+A6x3+3A6x2−2A6x−A6−A2x3−2A2x2+A2x+A−2x3−A−2x+A−2−A−6xS2,∞(517) = A2x2 − 3A−2x2 + A−2 + A−6x3 + 3A−6x2 − 2A−6x − A−6 − A−10x3 − A−10x2 + A−10x − A−10 +A−14x3 − A−14x + A−14 − A−18xS2,∞(518) = −2A−6x2+A−6+A−10x3+2A−10x2−2A−10x−2A−14x3−A−14x2+3A−14x+A−18x3+A−18−A−22xS2,∞(518) = A18x2−3A14x2+2A14+A10x3+2A10x2−2A10x−2A6x3+3A6x−A6+A2x3−A2x2+A2−A−2xS2,∞(519) = −A4x3 + 3x3 − 3x − 2A−4x3 + A−4x + 2A−8x3 − A−8x2 − 2A−8x + A−8 − A−12x3 + A−12x2 +A−16x − A−16

S2,∞(519) = A8x3−A8x−2A4x3+A4x+3x3−3x−2A−4x3+A−4x2+A−4x−A−4+A−8x3−A−8x2−A−12x+A−12

S2,∞(520) = 2A2x2 − A2 − A−2x3 − 3A−2x2 + 2A−2x + A−2 + 2A−6x3 + A−6x2 − 3A−6x + A−6 − 2A−10x3 +A−10x2 + 2A−10x − 2A−10 + A−14x3 − A−14x2 + A−14x − A−18x + A−18

S2,∞(520) = −A10x2 + 3A6x2 − A6 − 2A2x3 − 2A2x2 + 4A2x + 2A−2x3 − 2A−2x + A−2 − A−6x3 + A−6x2 −A−6 + A−10x3 − A−10x2 − A−14x + A−14

S2,∞(521) = A12x3 − 3A8x3 + 3A8x + 3A4x3 − 3A4x − 2x3 + x2 + 2x − 1 + A−4x3 − A−4x2 − A−8x + A−8

S2,∞(521) = −x3 + x + 3A−4x3 − 3A−4x − 3A−8x3 + 3A−8x + 2A−12x3 − A−12x2 − A−12x + A−12 − A−16x3 +A−16x2 + A−20x − A−20

S2,∞(522) = −A6x2 + 2A2x2 − A2 − A−2x3 − 2A−2x2 + 2A−2x + A−2 + A−6x3 − A−6x + 2A−6 − A−10xS2,∞(522) = A6x2 + A6 − 3A2x2 + 2A2 + A−2x3 + A−2x2 − 2A−2x − A−6x3 + A−6x − A−6 + A−10xS2,∞(524) = A−6 + A−10x3 − A−10x2 − 2A−10x + A−10 + A−14x2 − A−14x − A−14 − A−18

S2,∞(524) = −A18 + A14x2 − A14 + A6 + A2x3 − A2x2 − 2A2x + A2 − A−2xS2,∞(525) = −A−6x2+2A−10x2−4A−14x2+2A−14+A−18x3+3A−18x2−2A−18x−A−22x3−A−22x2+A−22x−A−22 + A−26x + A−26

S2,∞(525) = A26−2A22x2+4A18x2−A18−A14x3−3A14x2+2A14x+A14+A10x3+A10x2−A10x+A10−A6x2−A6xS2,∞(528) = A14x2 − 2A10x2 + 3A6x2 − A6 − 3A2x2 + A2 + A−2x3 + A−2x2 − 2A−2x + A−2 − A−6x − A−6

S2,∞(528) = −A6 + 2A2x2 − 3A−2x2 + A−2 + A−6x3 + 2A−6x2 − 2A−6x − 2A−10x2 − A−10x + A−14x2

S2,∞(534) = −A6x2+3A2x2−A2−3A−2x2+A−2+A−6x3+2A−6x2−2A−6x−A−10x2−A−10x−A−10+A−14

S2,∞(534) = A14 − A10x2 − A10 + 3A6x2 − A6 − 3A2x2 + A2 + A−2x3 + 2A−2x2 − 2A−2x − A−6x2 − A−6xS2,∞(535) = −A−6x2 + 2A−10x2 −A−10 − 3A−14x2 +A−18x3 + 3A−18x2 − 2A−18x −A−18 −A−22x3 −A−22x2 +A−22x − A−22 + A−26x + A−26

S2,∞(535) = A26−2A22x2+4A18x2−2A18−A14x3−2A14x2+2A14x−A14+A10x3+A10x2−A10x−A6x2−A6xS2,∞(538) = −A−2 + A−6x2 − A−6 − 2A−10x2 + A−14x3 + A−14x2 − 2A−14x − A−18x3 + A−18x + A−22xS2,∞(538) = −A18x2 + A18 + 2A14x2 − A14 − A10x3 − A10x2 + 2A10x − A10 + A6x3 − A6x − A2x − A2

S2,∞(541) = A18 + A10x2 − A10 − A6x2 + A2x3 − 2A2x − A−2xS2,∞(541) = A−2x2 − A−2 − A−6x2 + A−10x3 − 2A−10x − A−14x + A−18

S2,∞(543) = A6x2 −2A2x2 +A2 +A−2x3 +A−2x2 −2A−2x −A−6x3 +A−6x −A−6 +A−10x3 −A−10x −A−14xS2,∞(543) = A10x2−A10−A6x2+A2x3+A2x2−2A2x−A−2x3−A−2x2+A−2x+A−6x3−A−6x+A−6−A−10x


S2,∞(544) = −x3 + x + 3A−4x3 − 3A−4x − 3A−8x3 + 2A−8x + 3A−12x3 −A−12x2 − 3A−12x +A−12 −A−16x3 +A−16x2 − A−16x + A−20x − A−20

S2,∞(544) = −A16x +2A12x3−2A12x −3A8x3+2A8x +3A4x3−3A4x −2x3+x2+2x − 1+A−4x3−A−4x2−A−8x + A−8

S2,∞(545) = −A10x2 + 3A6x2 −A6 −A2x3 − 3A2x2 + 2A2x +A2 + 2A−2x3 − 3A−2x + 2A−2 −A−6x3 +A−10xS2,∞(545) = −A6x2 + A6 + 2A2x2 − A−2x3 − 3A−2x2 + 2A−2x + A−2 + 2A−6x3 + A−6x2 − 3A−6x + A−6 −A−10x3 − A−10 + A−14xS2,∞(546) = −x3 + x + 3A−4x3 − 2A−4x − 4A−8x3 + 4A−8x + 3A−12x3 −A−12x2 − 2A−12x +A−12 −A−16x3 +A−16x2 − A−16x + A−20x − A−20

S2,∞(546) = −A16x +2A12x3 −A12x −4A8x3 +4A8x + 3A4x3 −2A4x −2x3 + x2 +2x − 1+A−4x3 −A−4x2 −A−8x + A−8

S2,∞(547) = −A10x2+4A6x2−2A6−2A2x3−3A2x2+4A2x +3A−2x3−4A−2x +2A−2−2A−6x3+A−6x2+A−6x − A−6 + A−10x3 − A−10x2 + A−10x − A−14x + A−14

S2,∞(547) = −A6x2 + A6 + 3A2x2 − A2 − 2A−2x3 − 3A−2x2 + 4A−2x + 3A−6x3 + A−6x2 − 4A−6x + A−6 −2A−10x3 + A−10x2 + A−10x − 2A−10 + A−14x3 − A−14x2 + A−14x − A−18x + A−18

S2,∞(548) = A4x3−A4x−2x3+2x+3A−4x3−3A−4x−3A−8x3+A−8x2+3A−8x−A−8+A−12x3−A−12x2+A−12x − A−16x + A−16

S2,∞(548) = A12x − 2A8x3 + 2A8x + 3A4x3 − 3A4x − 2x3 + 2x + 2A−4x3 −A−4x2 − 2A−4x +A−4 −A−8x3 +A−8x2 + A−12x − A−12

S2,∞(549) = A14x2−2A10x2+A10+A6x3+2A6x2−2A6x−A2x3−A2x2+A2x−A2+A−2x3−A−2x−A−6xS2,∞(549) = A2x2−A2−2A−2x2+A−6x3+2A−6x2−2A−6x −A−10x3−A−10x2+A−10x +A−14x3−A−14x +A−14 − A−18xS2,∞(553) = A−2x2 − 2A−6x2 + A−6 + A−10x3 + A−10x2 − 2A−10x − A−14x2 − A−14x + A−18x2 − A−22

S2,∞(553) = −A22 + A18x2 − A14x2 + 2A10x2 − A10 − 2A6x2 + A6 + A2x3 − 2A2x + A2 − A−2xS2,∞(555) = −A18x2+3A14x2−A14−A10x3−3A10x2+2A10x+A10+A6x3+A6x2−A6x+A6−A2x2−A2x+A−2

S2,∞(555) = A2 − A−2x2 + 2A−6x2 − 4A−10x2 + 2A−10 + A−14x3 + 2A−14x2 − 2A−14x − A−18x3 + A−18x −A−18 + A−22xS2,∞(556) = A14x2 − 2A10x2 + A10 + 2A6x2 − 2A2x2 + A−2x3 + A−2x2 − 2A−2x − A−6x − A−6

S2,∞(556) = −A6+2A2x2−A2−2A−2x2+A−6x3+A−6x2−2A−6x+A−6−2A−10x2−A−10x+A−10+A−14x2

S2,∞(559) = A2x2 − 3A−2x2 + A−2 + A−6x3 + 3A−6x2 − 2A−6x − A−10x3 − 2A−10x2 + A−10x + A−14x3 +A−14x2 − A−14x − A−18x − A−18

S2,∞(559) = −A18 + 2A14x2 −A14 − 3A10x2 +A10 +A6x3 + 3A6x2 − 2A6x −A2x3 − 2A2x2 +A2x +A−2x3 −A−2x + A−2 − A−6xS2,∞(561) = A2x2−4A−2x2+A−2+A−6x3+4A−6x2−2A−6x −A−6−2A−10x3−2A−10x2+3A−10x −A−10+A−14x3 + A−14x2 − A−18x − A−18

S2,∞(561) = −A18 + 2A14x2 − A14 − 4A10x2 + A10 + A6x3 + 4A6x2 − 2A6x − A6 − 2A2x3 − 2A2x2 + 3A2x −A2 + A−2x3 + A−2 − A−6xS2,∞(562) = A8x − 2A4x3 + 5x3 − 4x − 4A−4x3 + 2A−4x + 3A−8x3 − A−8x2 − 2A−8x + A−8 − A−12x3 +A−12x2 − A−12x + A−16x − A−16

S2,∞(562) = −A12x + 2A8x3 −A8x − 4A4x3 + 2A4x + 5x3 − 4x − 3A−4x3 +A−4x2 +A−4x −A−4 +A−8x3 −A−8x2 + A−8x − A−12x + A−12

S2,∞(563) = −A6x2 + 3A2x2 − A2 − A−2x3 − 4A−2x2 + 2A−2x + A−2 + 2A−6x3 + 2A−6x2 − 3A−6x + A−6 −A−10x3 − A−10x2 + A−14x + A−14

S2,∞(563) = A14 − 2A10x2 + A10 + 4A6x2 − A6 − A2x3 − 4A2x2 + 2A2x + A2 + 2A−2x3 + A−2x2 − 3A−2x +A−2 − A−6x3 − A−6 + A−10x


S2,∞(564) = A8x −2A4x3+A4x +4x3−2x −4A−4x3+3A−4x +3A−8x3−A−8x2−2A−8x +A−8−A−12x3+A−12x2 − A−12x + A−16x − A−16

S2,∞(564) = −A12x + 2A8x3 −A8x −4A4x3 + 3A4x +4x3 − 2x − 3A−4x3 +A−4x2 + 2A−4x −A−4 +A−8x3 −A−8x2 + A−8x − A−12x + A−12

S2,∞(565) = −A6x2 + 5A2x2 − 2A2 − 2A−2x3 −6A−2x2 +4A−2x +A−2 +4A−6x3 + 3A−6x2 −6A−6x +A−6 −3A−10x3 + 2A−10x − 2A−10 + A−14x3 − A−14x2 + 2A−14x + A−14 − A−18x + A−18

S2,∞(565) = A14−3A10x2+A10+7A6x2−3A6−3A2x3−5A2x2+6A2x+4A−2x3+A−2x2−5A−2x+2A−2−2A−6x3 + A−6x2 − 2A−6 + A−10x3 − A−10x2 + A−10x − A−14x + A−14

S2,∞(566) = A10x2 − 2A6x2 + A6 + A2x3 + 2A2x2 − 2A2x − A−2x3 − 2A−2x2 + A−2x + A−6x3 + A−6x2 −A−6x − A−10x − A−10

S2,∞(566) = −A10+2A6x2−A6−3A2x2+A2+A−2x3+2A−2x2−2A−2x−A−6x3−A−6x2+A−6x+A−10x3−A−10x + A−10 − A−14xS2,∞(567) = −A−6x2 +A−6 +A−10x3 +A−10x2 − 2A−10x −A−14x3 +A−14x −A−14 +A−18x3 −A−18x −A−22xS2,∞(567) = A18x2 −A18 −A14x2 +A10x3 +A10x2 − 2A10x −A6x3 +A6x +A2x3 −A2x2 −A2x +A2 −A−2xS2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x − x3 + x + 2A−4x3 − A−4x2 − 3A−4x + A−4 − A−8x3 + A−8x2 +A−12x − A−12

S2,∞(568) = A4x3−2A4x − x3+ x +2A−4x3−3A−4x −2A−8x3+A−8x2+2A−8x −A−8+A−12x3−A−12x2−A−16x + A−16

S2,∞(569) = A−2x2 − A−2 − A−6x3 − A−6x2 + 2A−6x + 2A−10x3 − 3A−10x + A−10 − 2A−14x3 + A−14x2 +2A−14x − A−14 + A−18x3 − A−18x2 + A−18x − A−22x + A−22

S2,∞(569) = −A14x2 + A14 + 2A10x2 − A10 − 2A6x3 + 4A6x − A6 + 2A2x3 − A2x2 − 2A2x + A2 − A−2x3 +A−2x2 − A−2 + A−6x3 − A−6x2 − A−10x + A−10

S2,∞(570) = A12x3−A12x−2A8x3+2A8x+2A4x3−2A4x−2x3+x2+3x− 1+A−4x3−A−4x2−A−8x+A−8

S2,∞(570) = −x3+2x +2A−4x3−2A−4x −2A−8x3+2A−8x +2A−12x3 −A−12x2 −2A−12x +A−12 −A−16x3 +A−16x2 + A−20x − A−20

S2,∞(571) = −A6x3 + 2A6x + A2x3 − A2x2 − A2x + A−2x2 − A−2x − A−2 − A−6

S2,∞(571) = −A6 +A2x2 −A2 −A−2 −A−6x2 +A−6 +A−10x3 −A−10x2 − 2A−10x +A−10 −A−14x3 +A−14x2 +A−14x − A−14 + A−18xS2,∞(572) = A−8x3 − A−8x − 2A−12x3 + 3A−12x + 2A−16x3 − A−16x2 − 2A−16x + A−16 − A−20x3 + A−20x2 +A−24x − A−24

S2,∞(572) = A16x3−A16x −2A12x3+3A12x +A8x3−A8x −A4x3+A4x2+A4x −A4+ x3− x2−A−4x +A−4

S2,∞(573) = A6x2−A2x3−A2x2+2A2x+A−2x3−A−2x−A−6x3+A−6x2+A−6x−A−6+A−10x3−A−10x2−A−14x + A−14

S2,∞(573) = A2x2−A2−A−2x3−A−2x2+2A−2x +A−6x3−A−6x +A−6−A−10x3+A−10x2+A−10x −A−10+A−14x3 − A−14x2 − A−18x + A−18

S2,∞(574) = A−14x3 − A−14x2 − 2A−14x + A−14 + A−18x2 − A−18x − A−22

S2,∞(574) = −A22 + A18x2 + A6x3 − A6x2 − 2A6x + A6 − A2xS2,∞(575) = A2−A−6x2+A−6+A−10x3−A−10x2−2A−10x+A−10−A−14x3+A−14x2+A−14x−A−14+A−18xS2,∞(577) = −A6x2 + A6 + 2A2x2 − A2 − A−2x2 + A−6x3 − 2A−6x + A−6 − A−10x − A−10

S2,∞(577) = −A10 + A6x2 − A2x2 + A−2x3 + A−2x2 − 2A−2x − A−6x2 − A−6x + A−6

S2,∞(579) = −A−6x2+A−6+A−10x2−2A−14x2+A−14+A−18x3+A−18x2−2A−18x+A−18−A−22x3+A−22x−A−22 + A−26xS2,∞(579) = −A22x2 + 2A18x2 − A14x3 − A14x2 + 2A14x + A10x3 − A10x + A10 − A6x2 − A6x + A6

S2,∞(580) = A14x2 − A14 − A10x2 + A6x2 − A2x2 + A−2x3 − 2A−2x + A−2 − A−6xS2,∞(580) = A2x2 − A−2x2 + A−6x3 − 2A−6x + A−6 − A−10x2 − A−10x + A−14x2 − A−14


S2,∞(581) = A−14x3 − A−14x2 − 2A−14x + 2A−18x2 − A−18x − A−18 − A−22x2 + A−26

S2,∞(581) = A26 − A22x2 + 2A18x2 − A18 − A14 + A6x3 − A6x2 − 2A6x + A6 − A2xS2,∞(582) = −A−6x2 + A−10x3 + 2A−10x2 − 2A−10x − A−10 − A−14x3 − A−14x2 + A−14x + A−18x3 − A−18x +A−18 − A−22xS2,∞(582) = A18x2 − 2A14x2 + A14 + A10x3 + 2A10x2 − 2A10x − A10 − A6x3 + A6x − A6 + A2x3 − A2x2 −A2x + A2 − A−2xS2,∞(583) = −A4x3+A4x+2x3−x−2A−4x3+2A−4x+2A−8x3−A−8x2−2A−8x+A−8−A−12x3+A−12x2+A−16x − A−16

S2,∞(583) = A8x3 − A8x − 2A4x3 + 2A4x + 2x3 − x − 2A−4x3 + A−4x2 + 2A−4x − A−4 + A−8x3 − A−8x2 −A−12x + A−12

S2,∞(584) = A−2x2 − A−6x3 − 2A−6x2 + 2A−6x + A−6 + 2A−10x3 + A−10x2 − 3A−10x − 2A−14x3 + A−14x2 +2A−14x − 2A−14 + A−18x3 − A−18x2 + A−18x − A−22x + A−22

S2,∞(584) = −A14x2 + 3A10x2 − 2A10 − 2A6x3 − A6x2 + 4A6x + 2A2x3 − A2x2 − 2A2x + 2A2 − A−2x3 +A−2x2 − A−2 + A−6x3 − A−6x2 − A−10x + A−10

S2,∞(585) = −A−18x3 + A−18x2 + 2A−18x − A−18 + A−22x3 − A−22x2 − A−26x + A−26

S2,∞(585) = −A10x3 + A10x2 + 2A10x − A10 + A6x + A−2x3 − A−2x2 − A−2x − A−6x + A−6

Knots in L(8, 1)

S2,∞(42) = −A8x3 + A8x + A4x3 − A4x + A2x3 − 2A2x − A−2xS2,∞(42) = −x3 + 2x + A−4x3 − A−4x + A−6x3 − 2A−6x − A−8x − A−10xS2,∞(45) = −A4x3 + 2x3 − x + A−2x3 − 2A−2x − A−4x3 − A−6x + A−8xS2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x + A−2x3 − 2A−2x − 2A−4x − A−6xS2,∞(414) = A8x − 2A4x3 + A4x + 3x3 − 2x + A−2x3 − 2A−2x − A−4x3 − A−4x − A−6x + A−8xS2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x + A−6x3 − 2A−6x − A−8x3 − A−10x + A−12xS2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x + A2x3 − 2A2x − x − A−2xS2,∞(421) = −A4x3 + A4x + x3 + A−2x3 − 2A−2x − A−4x − A−6xS2,∞(424) = −A12x3 + 2A12x + A6x3 − 2A6x + A4x3 − 2A4x − A2xS2,∞(55) = A−8x3 − 2A−8x + A−10x3 − 2A−10x − A−12x − A−14x3 + A−14x + A−18xS2,∞(55) = A16x3 − 2A16x − A12x3 + A12x − A10x3 + 2A10x + A6x3 − A6x + A4x3 − 2A4x − A2xS2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 − A2x2 + A2 + A−2

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 + A−14

S2,∞(58) = −x3 + x +3A−4x3 −4A−4x +A−6x3 −2A−6x −A−8x3 −A−8x −A−10x3 +A−10x +A−12x +A−14xS2,∞(58) = A12x3 − A12x − 2A8x3 + A8x − A6x3 + 2A6x + 2A4x3 − 2A4x + A2x3 − A2x − x − A−2xS2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 + A−2x4 − 2A−2x2 + A−2 − A−6x2 + 2A−6

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 + A−6x2 + A−10

S2,∞(512) = −x3+4A−4x3−4A−4x+A−6x3−2A−6x−3A−8x3+A−8x−A−10x3+A−10x+A−12x3+A−12x+A−14x − A−16xS2,∞(512) = −A16x+2A12x3−A12x−4A8x3+3A8x−A6x3+2A6x+3A4x3−2A4x+A2x3−A2x−2x−A−2xS2,∞(514) = −A4x3 + 3x3 − 2x + A−2x3 − 2A−2x − 2A−4x3 − A−6x3 + A−6x + A−8x3 + A−10x − A−12xS2,∞(514) = −A12x + 2A8x3 − 2A8x − 3A4x3 + 2A4x −A2x3 + 2A2x + 2x3 +A−2x3 −A−2x − 2A−4x −A−6xS2,∞(515) = −A10x2 + 2A6x4 − 2A6x2 + A6 − 3A2x4 + 3A2x2 + A−2x4 − A−6x2 + A−6

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 + A−6x4 + A−6 − A−10x2 + A−10


S2,∞(516) = A8x3−3A4x3+2A4x −A2x3+2A2x +3x3− x +A−2x3−A−2x −A−4x3−A−4x −A−6x +A−8xS2,∞(516) = A8x − 2A4x3 + A4x + 4x3 − 3x + A−2x3 − 2A−2x − 2A−4x3 − A−6x3 + A−6x + 2A−8x + A−10xS2,∞(519) = −A4x3 + 3x3 − 3x + A−2x3 − 2A−2x − A−4x3 − A−4x − A−6x3 + A−6x + A−8x + A−10xS2,∞(519) = A8x3 − A8x − 2A4x3 + A4x − A2x3 + 2A2x + 2x3 − x + A−2x3 − A−2x − 2A−4x − A−6xS2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 + A−6x4 − A−6x2 + A−6 − A−10x2 + A−10

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 + A6 − 2A2x4 + 2A2x2 + A−2x2 + A−6

S2,∞(521) = A12x3 − 3A8x3 + 3A8x − A6x3 + 2A6x + 2A4x3 − A4x + A2x3 − A2x − x − A−2xS2,∞(521) = −x3+x+3A−4x3−3A−4x+A−6x3−2A−6x−2A−8x3+A−8x−A−10x3+A−10x+2A−12x+A−14xS2,∞(544) = −x3+x+3A−4x3−3A−4x+A−6x3−2A−6x−2A−8x3−A−10x3+A−10x+A−12x3+A−14x−A−16xS2,∞(544) = −A16x+2A12x3−2A12x−3A8x3+2A8x−A6x3+2A6x+2A4x3−A4x+A2x3−A2x−x−A−2xS2,∞(546) = −x3 + x + 3A−4x3 − 2A−4x + A−6x3 − 2A−6x − 3A−8x3 + 2A−8x − A−10x3 + A−10x + A−12x3 +A−12x + A−14x − A−16xS2,∞(546) = −A16x + 2A12x3 − A12x − 4A8x3 + 4A8x − A6x3 + 2A6x + 2A4x3 + A2x3 − A2x − x − A−2xS2,∞(547) = −A10x2 + 2A6x4 − 2A6x2 − 3A2x4 + 4A2x2 − A2 + A−2x4 − A−2x2 + A−2 − A−6x2 + 2A−6

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 + A−6x4 − A−10x2 + A−10

S2,∞(548) = A4x3−A4x−2x3+2x−A−2x3+2A−2x+2A−4x3−A−4x+A−6x3−A−6x−A−8x3−A−10x+A−12xS2,∞(548) = A12x − 2A8x3 + 2A8x + 3A4x3 − 3A4x + A2x3 − 2A2x − x3 − A−2x3 + A−2x + A−4x + A−6xS2,∞(562) = A8x−2A4x3+5x3−4x+A−2x3−2A−2x−3A−4x3−A−6x3+A−6x+A−8x3+A−8x+A−10x−A−12xS2,∞(562) = −A12x + 2A8x3 − A8x − 4A4x3 + 2A4x − A2x3 + 2A2x + 4x3 − 2x + A−2x3 − A−2x − A−4x3 −2A−4x − A−6x + A−8xS2,∞(564) = A8x − 2A4x3 +A4x + 4x3 − 2x +A−2x3 − 2A−2x − 3A−4x3 +A−4x −A−6x3 +A−6x +A−8x3 +A−8x + A−10x − A−12xS2,∞(564) = −A12x +2A8x3 −A8x −4A4x3 + 3A4x −A2x3 +2A2x + 3x3 +A−2x3 −A−2x −A−4x3 −A−4x −A−6x + A−8xS2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 + 2A−6x4 − 3A−10x2 + 2A−10 + A−14

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 + A−2x4 + A−2x2 − A−6x2 + A−6

S2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x + A2x3 − 2A2x − x − A−2x3 + A−2x + A−6xS2,∞(568) = A4x3 − 2A4x − x3 + x − A−2x3 + 2A−2x + A−4x3 − A−4x + A−6x3 − A−6x − A−8x − A−10xS2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 − A−6 + A−10x4 − 2A−10x2 + A−10 − A−14x2 + 2A−14

S2,∞(569) = −A14x2 + A14 + 2A10x4 − 4A10x2 + A10 − 2A6x4 + 4A6x2 − A6 + A−2

S2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x − A6x3 + 2A6x + A4x3 + A2x3 − A2x − A−2xS2,∞(570) = −x3 + 2x + 2A−4x3 − 2A−4x + A−6x3 − 2A−6x − A−8x3 − A−10x3 + A−10x + A−12x + A−14xS2,∞(572) = A−8x3 − A−8x + A−10x3 − 2A−10x − A−12x3 + A−12x − A−14x3 + A−14x + A−16x + A−18xS2,∞(572) = A16x3 − A16x − 2A12x3 + 3A12x − A10x3 + 2A10x + A8x + A6x3 − A6x + A4x3 − 2A4x − A2xS2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 + A−6

S2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 + A−6 + A−10

S2,∞(583) = −A4x3 + A4x + 2x3 − x + A−2x3 − 2A−2x − A−4x3 − A−6x3 + A−6x + A−8x + A−10xS2,∞(583) = A8x3 − A8x − 2A4x3 + 2A4x − A2x3 + 2A2x + x3 + x + A−2x3 − A−2x − A−4x − A−6xS2,∞(584) = A−2x4 − 2A−2x2 + A−2 − 2A−6x4 + 3A−6x2 + A−10x4 − A−10x2 − A−14x2 + A−14

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 + A2 + A−2

S2,∞(585) = A14x4 − 3A14x2 + A14 − A6x4 + 2A6x2 + A2


Knots in L(8, 3)

S2,∞(42) = −A8x3 + A8x + A4x3 − A4x − x3 + 2x + A−2x3 − 2A−2x + A−4x3 − 2A−4x − A−6xS2,∞(42) = −x3 + 2x + A−4x3 − A−4x − A−8x3 + A−8x + A−10x3 − 2A−10x + A−12x3 − 2A−12x − A−14xS2,∞(45) = −A4x3 + 2x3 − x − 2A−4x3 + 2A−4x + A−6x3 − 2A−6x + A−8x3 − A−8x − A−10xS2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x − A−4x3 + A−6x3 − 2A−6x + A−8x3 − 2A−8x − A−10xS2,∞(414) = A8x − 2A4x3 + A4x + 3x3 − 2x − 2A−4x3 + A−4x + A−6x3 − 2A−6x + A−8x3 − A−8x − A−10xS2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x − 2A−8x3 + 2A−8x + A−10x3 − 2A−10x + A−12x3 − A−12x − A−14xS2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x − x3 + x + A−2x3 − 2A−2x + A−4x3 − 2A−4x − A−6xS2,∞(421) = −A4x3 + A4x + x3 − A−4x3 + A−4x + A−6x3 − 2A−6x + A−8x3 − 2A−8x − A−10xS2,∞(424) = −A−12x3 + 2A−12x + A−14x3 − 2A−14x + A−16x3 − 2A−16x − A−18xS2,∞(424) = −A12x3 + 2A12x + A2x3 − 2A2x + x3 − 2x − A−2xS2,∞(55) = A−8x3 − 2A−8x − A−12x3 + A−12x + A−14x3 − 2A−14x + 2A−16x3 − 4A−16x − A−18x3 + A−18x −A−20x3 + 2A−20x + A−22xS2,∞(55) = A16x3 − 2A16x − A12x3 + A12x + A8x3 − 2A8x − A6x3 + 2A6x − A4x3 + 2A4x + A2x3 − A2x +x3 − 2x − A−2xS2,∞(56) = A10x4−2A10x2−A6x4+2A6x2+A2x4−3A2x2+A2−A−2x4+3A−2x2−A−2−A−10x2+2A−10

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 + A−10x4 − 2A−10x2 − A−14x4 + 3A−14x2 − A−14 −A−22x2 + 2A−22

S2,∞(58) = −x3 + x + 3A−4x3 −4A−4x − 2A−8x3 +A−8x +A−10x3 − 2A−10x + 2A−12x3 − 3A−12x −A−14x3 +A−14x − A−16x3 + 2A−16x + A−18xS2,∞(58) = A12x3 − A12x − 2A8x3 + A8x + 3A4x3 − 4A4x − A2x3 + 2A2x − 2x3 + 3x + A−2x3 − A−2x +A−4x3 − 2A−4x − A−6xS2,∞(59) = A6x4−A6x2−2A2x4+3A2x2−A2+2A−2x4−4A−2x2+A−2−A−6x4+2A−6x2−A−14x2+2A−14

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 + A−6x4 − A−6x2 − A−10x4 +3A−10x2 − A−10 − A−18x2 + 2A−18

S2,∞(512) = −x3 + 4A−4x3 − 4A−4x − 4A−8x3 + 3A−8x + A−10x3 − 2A−10x + 3A−12x3 − 3A−12x − A−14x3 +A−14x − A−16x3 + A−16x + A−18xS2,∞(512) = −A16x + 2A12x3 −A12x − 4A8x3 + 3A8x + 4A4x3 − 4A4x −A2x3 + 2A2x − 2x3 + 2x +A−2x3 −A−2x + A−4x3 − 2A−4x − A−6xS2,∞(514) = −A4x3 + 3x3 − 2x − 3A−4x3 + 2A−4x +A−6x3 − 2A−6x + 3A−8x3 − 4A−8x −A−10x3 +A−10x −A−12x3 + A−12x + A−14xS2,∞(514) = −A12x + 2A8x3 − 2A8x − 3A4x3 + 2A4x + 3x3 − 2x − A−2x3 + 2A−2x − 2A−4x3 + 2A−4x +A−6x3 − A−6x + A−8x3 − 2A−8x − A−10xS2,∞(515) = −A10x2 +2A6x4 −2A6x2 +A6 − 3A2x4 + 3A2x2 +2A−2x4 −2A−2x2 −A−6x4 +2A−6x2 −A−6 −A−14x2 + 2A−14

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 + 2A−6x4 − 2A−6x2 + A−6 − A−10x4 + 2A−10x2 −A−10 − A−18x2 + 2A−18

S2,∞(516) = A8x3−3A4x3+2A4x+4x3−3x−A−2x3+2A−2x−3A−4x3+3A−4x+A−6x3−A−6x+A−8x3−A−8x − A−10xS2,∞(516) = A8x − 2A4x3 + A4x + 4x3 − 3x − 3A−4x3 + 2A−4x + A−6x3 − 2A−6x + 2A−8x3 − 2A−8x −A−10x3 + A−10x − A−12x3 + 2A−12x + A−14xS2,∞(519) = −A4x3 + 3x3 − 3x − 2A−4x3 + A−4x + A−6x3 − 2A−6x + 2A−8x3 − 3A−8x − A−10x3 + A−10x −A−12x3 + 2A−12x + A−14xS2,∞(519) = A8x3 −A8x − 2A4x3 +A4x + 3x3 − 3x −A−2x3 + 2A−2x − 2A−4x3 + 2A−4x +A−6x3 −A−6x +


A−8x3 − 2A−8x − A−10xS2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 + 2A−6x4 − 3A−6x2 + A−6 − A−10x4 + 2A−10x2 − A−10 −A−18x2 + 2A−18

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 + A6 − 2A2x4 + 2A2x2 + A−2x4 − A−2x2 − A−6x4 + 3A−6x2 − A−6 −A−14x2 + 2A−14

S2,∞(521) = A12x3 − 3A8x3 + 3A8x + 3A4x3 − 3A4x − A2x3 + 2A2x − 2x3 + 3x + A−2x3 − A−2x + A−4x3 −2A−4x − A−6xS2,∞(521) = −x3+x+3A−4x3−3A−4x−3A−8x3+3A−8x+A−10x3−2A−10x+2A−12x3−2A−12x−A−14x3+A−14x − A−16x3 + 2A−16x + A−18xS2,∞(544) = −x3+x+3A−4x3−3A−4x−3A−8x3+2A−8x+A−10x3−2A−10x+3A−12x3−4A−12x−A−14x3+A−14x − A−16x3 + A−16x + A−18xS2,∞(544) = −A16x + 2A12x3 − 2A12x − 3A8x3 + 2A8x + 3A4x3 − 3A4x −A2x3 + 2A2x − 2x3 + 3x +A−2x3 −A−2x + A−4x3 − 2A−4x − A−6xS2,∞(546) = −x3+x+3A−4x3−2A−4x−4A−8x3+4A−8x+A−10x3−2A−10x+3A−12x3−3A−12x−A−14x3+A−14x − A−16x3 + A−16x + A−18xS2,∞(546) = −A16x + 2A12x3 −A12x − 4A8x3 + 4A8x + 3A4x3 − 2A4x −A2x3 + 2A2x − 2x3 + 3x +A−2x3 −A−2x + A−4x3 − 2A−4x − A−6xS2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2+2A−2x4−3A−2x2+A−2−A−6x4+2A−6x2−A−14x2 + 2A−14

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 + 2A−6x4 − 2A−6x2 − A−10x4 +2A−10x2 − A−10 − A−18x2 + 2A−18

S2,∞(548) = A4x3 − A4x − 2x3 + 2x + 3A−4x3 − 3A−4x − A−6x3 + 2A−6x − 3A−8x3 + 4A−8x + A−10x3 −A−10x + A−12x3 − A−12x − A−14xS2,∞(548) = A12x−2A8x3+2A8x+3A4x3−3A4x−2x3+2x+A−2x3−2A−2x+2A−4x3−3A−4x−A−6x3+A−6x − A−8x3 + 2A−8x + A−10xS2,∞(562) = A8x − 2A4x3 + 5x3 − 4x − 4A−4x3 + 2A−4x + A−6x3 − 2A−6x + 3A−8x3 − 3A−8x − A−10x3 +A−10x − A−12x3 + A−12x + A−14xS2,∞(562) = −A12x+2A8x3−A8x−4A4x3+2A4x+5x3−4x−A−2x3+2A−2x−3A−4x3+2A−4x+A−6x3−A−6x + A−8x3 − A−8x − A−10xS2,∞(564) = A8x − 2A4x3 + A4x + 4x3 − 2x − 4A−4x3 + 3A−4x + A−6x3 − 2A−6x + 3A−8x3 − 3A−8x −A−10x3 + A−10x − A−12x3 + A−12x + A−14xS2,∞(564) = −A12x+2A8x3−A8x−4A4x3+3A4x+4x3−2x−A−2x3+2A−2x−3A−4x3+3A−4x+A−6x3−A−6x + A−8x3 − A−8x − A−10xS2,∞(565) = −A6x2+2A2x4−A2x2−4A−2x4+4A−2x2−A−2+3A−6x4−2A−6x2−A−10x4+A−14−A−18x2+2A−18

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 + 2A−2x4 − A−2x2 − A−6x4 +2A−6x2 − A−6 − A−14x2 + 2A−14

S2,∞(568) = −A8x3 +A8x + 2A4x3 − 3A4x − x3 + x +A−2x3 − 2A−2x + 2A−4x3 − 4A−4x −A−6x3 +A−6x −A−8x3 + 2A−8x + A−10xS2,∞(568) = A4x3 − 2A4x − x3 + x + 2A−4x3 − 3A−4x − A−6x3 + 2A−6x − 2A−8x3 + 3A−8x + A−10x3 −A−10x + A−12x3 − 2A−12x − A−14xS2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 −A−6 + 2A−10x4 − 4A−10x2 +A−10 −A−14x4 + 2A−14x2 −A−22x2 + 2A−22

S2,∞(569) = −A14x2 + A14 + 2A10x4 − 4A10x2 + A10 − 2A6x4 + 4A6x2 − A6 + A2x4 − 2A2x2 − A−2x4 +3A−2x2 − A−2 − A−10x2 + 2A−10


S2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x − A2x3 + 2A2x − 2x3 + 4x + A−2x3 − A−2x +A−4x3 − 2A−4x − A−6xS2,∞(570) = −x3+2x+2A−4x3−2A−4x−2A−8x3+2A−8x+A−10x3−2A−10x+2A−12x3−3A−12x−A−14x3+A−14x − A−16x3 + 2A−16x + A−18xS2,∞(572) = A−8x3 −A−8x − 2A−12x3 + 3A−12x +A−14x3 − 2A−14x + 2A−16x3 − 3A−16x −A−18x3 +A−18x −A−20x3 + 2A−20x + A−22xS2,∞(572) = A16x3 − A16x − 2A12x3 + 3A12x + A8x3 − A8x − A6x3 + 2A6x − A4x3 + 2A4x + A2x3 − A2x +x3 − 2x − A−2xS2,∞(573) = A6x4−2A6x2+A6−A2x4+A2x2+A−2x4−2A−2x2−A−6x4+3A−6x2−A−6−A−14x2+2A−14

S2,∞(573) = A2x4−2A2x2−A−2x4+A−2x2+A−6x4−2A−6x2+A−6−A−10x4+3A−10x2−A−10−A−18x2+2A−18

S2,∞(583) = −A4x3 + A4x + 2x3 − x − 2A−4x3 + 2A−4x + A−6x3 − 2A−6x + 2A−8x3 − 3A−8x − A−10x3 +A−10x − A−12x3 + 2A−12x + A−14xS2,∞(583) = A8x3 −A8x − 2A4x3 + 2A4x + 2x3 − x −A−2x3 + 2A−2x − 2A−4x3 + 3A−4x +A−6x3 −A−6x +A−8x3 − 2A−8x − A−10xS2,∞(584) = A−2x4 − 2A−2x2 +A−2 − 2A−6x4 + 3A−6x2 + 2A−10x4 − 3A−10x2 −A−14x4 + 2A−14x2 −A−14 −A−22x2 + 2A−22

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 + A2x4 − 2A2x2 + A2 − A−2x4 + 3A−2x2 − A−2 −A−10x2 + 2A−10

S2,∞(585) = A−14x4 − 3A−14x2 + A−14 − A−18x4 + 3A−18x2 − A−18 − A−26x2 + 2A−26

S2,∞(585) = A14x4 − 3A14x2 + A14 − A2x4 + 3A2x2 − A2 − A−6x2 + 2A−6

Knots in L(9, 1)

S2,∞(42) = −A8x3 + A8x + A4x3 − A4x + A3x4 − 3A3x2 + A3 − A−1x2 + A−1

S2,∞(42) = −x3 + 2x + A−4x3 − A−4x + A−5x4 − 3A−5x2 + A−5 − A−8x − A−9x2 + A−9

S2,∞(45) = −A4x3 + 2x3 − x + A−1x4 − 3A−1x2 + A−1 − A−4x3 − A−5x2 + A−5 + A−8xS2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x + A−1x4 − 3A−1x2 + A−1 − 2A−4x − A−5x2 + A−5

S2,∞(414) = A8x − 2A4x3 +A4x + 3x3 − 2x +A−1x4 − 3A−1x2 +A−1 −A−4x3 −A−4x −A−5x2 +A−5 +A−8xS2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x + A−5x4 − 3A−5x2 + A−5 − A−8x3 − A−9x2 + A−9 + A−12xS2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x + A3x4 − 3A3x2 + A3 − x − A−1x2 + A−1

S2,∞(421) = −A4x3 + A4x + x3 + A−1x4 − 3A−1x2 + A−1 − A−4x − A−5x2 + A−5

S2,∞(424) = −A12x3 + 2A12x + A7x4 − 3A7x2 + A7 + A4x3 − 2A4x − A3x2 + A3

S2,∞(55) = A−8x3 − 2A−8x + A−9x4 − 3A−9x2 + A−9 − A−12x − A−13x4 + 2A−13x2 + A−17x2 − A−17

S2,∞(55) = A16x3−2A16x −A12x3+A12x −A11x4+3A11x2−A11+A7x4−2A7x2+A4x3−2A4x −A3x2+A3

S2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 − A3x3 + 2A3x + A−1xS2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 − A−9x3 + 2A−9x + A−10x2 − A−10 + A−13xS2,∞(58) = −x3+x+3A−4x3−4A−4x+A−5x4−3A−5x2+A−5−A−8x3−A−8x−A−9x4+2A−9x2+A−12x+A−13x2 − A−13

S2,∞(58) = A12x3−A12x−2A8x3+A8x−A7x4+3A7x2−A7+2A4x3−2A4x+A3x4−2A3x2−x−A−1x2+A−1

S2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 − A−1x3 + 2A−1x + A−2x4 − A−2x2 + A−5x − A−6x2 + A−6

S2,∞(59) = −A6x2+A6+2A2x4−4A2x2+A2−2A−2x4+3A−2x2−A−2−A−5x3+2A−5x+2A−6x2−A−6+A−9xS2,∞(512) = −x3+4A−4x3−4A−4x+A−5x4−3A−5x2+A−5−3A−8x3+A−8x−A−9x4+2A−9x2+A−12x3+A−12x + A−13x2 − A−13 − A−16x


S2,∞(512) = −A16x +2A12x3−A12x −4A8x3+3A8x −A7x4+3A7x2−A7+3A4x3−2A4x +A3x4−2A3x2−2x − A−1x2 + A−1

S2,∞(514) = −A4x3+3x3−2x+A−1x4−3A−1x2+A−1−2A−4x3−A−5x4+2A−5x2+A−8x3+A−9x2−A−9−A−12xS2,∞(514) = −A12x+2A8x3−2A8x−3A4x3+2A4x−A3x4+3A3x2−A3+2x3+A−1x4−2A−1x2−2A−4x−A−5x2 + A−5

S2,∞(515) = −A10x2 + 2A6x4 − 2A6x2 + A6 − 3A2x4 + 3A2x2 − A−1x3 + 2A−1x + A−2x4 + A−2x2 − A−2 +A−5x − A−6x2

S2,∞(515) = −A6x2+2A2x4−2A2x2−3A−2x4+3A−2x2−A−5x3+2A−5x+A−6x4+A−6x2+A−9x−A−10x2

S2,∞(516) = A8x3 − 3A4x3 + 2A4x − A3x4 + 3A3x2 − A3 + 3x3 − x + A−1x4 − 2A−1x2 − A−4x3 − A−4x −A−5x2 + A−5 + A−8xS2,∞(516) = A8x − 2A4x3 +A4x +4x3 − 3x +A−1x4 − 3A−1x2 +A−1 − 2A−4x3 −A−5x4 + 2A−5x2 + 2A−8x +A−9x2 − A−9

S2,∞(519) = −A4x3+3x3−3x+A−1x4−3A−1x2+A−1−A−4x3−A−4x−A−5x4+2A−5x2+A−8x+A−9x2−A−9

S2,∞(519) = A8x3−A8x−2A4x3+A4x−A3x4+3A3x2−A3+2x3−x+A−1x4−2A−1x2−2A−4x−A−5x2+A−5

S2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 − A−5x3 + 2A−5x + A−6x4 + A−9x − A−10x2

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 + A6 − 2A2x4 + 2A2x2 − A−1x3 + 2A−1x + 2A−2x2 − A−2 + A−5xS2,∞(521) = A12x3 − 3A8x3 + 3A8x −A7x4 + 3A7x2 −A7 + 2A4x3 −A4x +A3x4 − 2A3x2 − x −A−1x2 +A−1

S2,∞(521) = −x3 + x + 3A−4x3 − 3A−4x + A−5x4 − 3A−5x2 + A−5 − 2A−8x3 + A−8x − A−9x4 + 2A−9x2 +2A−12x + A−13x2 − A−13

S2,∞(544) = −x3 + x + 3A−4x3 − 3A−4x + A−5x4 − 3A−5x2 + A−5 − 2A−8x3 − A−9x4 + 2A−9x2 + A−12x3 +A−13x2 − A−13 − A−16xS2,∞(544) = −A16x+2A12x3−2A12x−3A8x3+2A8x−A7x4+3A7x2−A7+2A4x3−A4x+A3x4−2A3x2−x − A−1x2 + A−1

S2,∞(546) = −x3 + x + 3A−4x3 − 2A−4x + A−5x4 − 3A−5x2 + A−5 − 3A−8x3 + 2A−8x − A−9x4 + 2A−9x2 +A−12x3 + A−12x + A−13x2 − A−13 − A−16xS2,∞(546) = −A16x + 2A12x3 − A12x − 4A8x3 + 4A8x − A7x4 + 3A7x2 − A7 + 2A4x3 + A3x4 − 2A3x2 − x −A−1x2 + A−1

S2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2−A−1x3+2A−1x+A−2x4+A−5x−A−6x2+A−6

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 − A−5x3 + 2A−5x + A−6x4 +A−6x2 − A−6 + A−9x − A−10x2

S2,∞(548) = A4x3 −A4x − 2x3 + 2x −A−1x4 + 3A−1x2 −A−1 + 2A−4x3 −A−4x +A−5x4 − 2A−5x2 −A−8x3 −A−9x2 + A−9 + A−12xS2,∞(548) = A12x − 2A8x3 + 2A8x + 3A4x3 − 3A4x + A3x4 − 3A3x2 + A3 − x3 − A−1x4 + 2A−1x2 + A−4x +A−5x2 − A−5

S2,∞(562) = A8x −2A4x3+5x3−4x +A−1x4−3A−1x2+A−1−3A−4x3−A−5x4+2A−5x2+A−8x3+A−8x +A−9x2 − A−9 − A−12xS2,∞(562) = −A12x + 2A8x3 − A8x − 4A4x3 + 2A4x − A3x4 + 3A3x2 − A3 + 4x3 − 2x + A−1x4 − 2A−1x2 −A−4x3 − 2A−4x − A−5x2 + A−5 + A−8xS2,∞(564) = A8x − 2A4x3 +A4x + 4x3 − 2x +A−1x4 − 3A−1x2 +A−1 − 3A−4x3 +A−4x −A−5x4 + 2A−5x2 +A−8x3 + A−8x + A−9x2 − A−9 − A−12xS2,∞(564) = −A12x +2A8x3 −A8x −4A4x3 + 3A4x −A3x4 + 3A3x2 −A3 + 3x3 +A−1x4 −2A−1x2 −A−4x3 −A−4x − A−5x2 + A−5 + A−8xS2,∞(565) = −A6x2 + 2A2x4 −A2x2 −4A−2x4 +4A−2x2 −A−2 −A−5x3 + 2A−5x + 2A−6x4 +A−6x2 −A−6 +A−9x − 3A−10x2 + A−10 + A−14

S2,∞(565) = A14−3A10x2+A10+3A6x4−2A6x2−4A2x4+4A2x2−A2−A−1x3+2A−1x+A−2x4+2A−2x2−


A−2 + A−5x − A−6x2

S2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x + A3x4 − 3A3x2 + A3 − x − A−1x4 + 2A−1x2 + A−5x2 − A−5

S2,∞(568) = A4x3−2A4x−x3+x−A−1x4+3A−1x2−A−1+A−4x3−A−4x+A−5x4−2A−5x2−A−8x−A−9x2+A−9

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 − A−6 − A−9x3 + 2A−9x + A−10x4 − A−10x2 + A−13x −A−14x2 + A−14

S2,∞(569) = −A14x2+A14+2A10x4−4A10x2+A10−2A6x4+4A6x2−A6−A3x3+2A3x+A2x2−A2+A−1xS2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x − A7x4 + 3A7x2 − A7 + A4x3 + A3x4 − 2A3x2 − A−1x2 + A−1

S2,∞(570) = −x3 + 2x + 2A−4x3 − 2A−4x + A−5x4 − 3A−5x2 + A−5 − A−8x3 − A−9x4 + 2A−9x2 + A−12x +A−13x2 − A−13

S2,∞(572) = A−8x3−A−8x+A−9x4−3A−9x2+A−9−A−12x3+A−12x−A−13x4+2A−13x2+A−16x+A−17x2−A−17

S2,∞(572) = A16x3 −A16x − 2A12x3 + 3A12x −A11x4 + 3A11x2 −A11 +A8x +A7x4 − 2A7x2 +A4x3 − 2A4x −A3x2 + A3

S2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 − A−1x3 + 2A−1x + A−2x2 − A−2 + A−5xS2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 − A−5x3 + 2A−5x + A−6x2 + A−9xS2,∞(583) = −A4x3+A4x+2x3−x+A−1x4−3A−1x2+A−1−A−4x3−A−5x4+2A−5x2+A−8x+A−9x2−A−9

S2,∞(583) = A8x3−A8x−2A4x3+2A4x−A3x4+3A3x2−A3+x3+x+A−1x4−2A−1x2−A−4x−A−5x2+A−5

S2,∞(584) = A−2x4−2A−2x2+A−2−2A−6x4+3A−6x2−A−9x3+2A−9x +A−10x4−A−10+A−13x −A−14x2

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 − A3x3 + 2A3x + A2x2 + A−1xS2,∞(585) = A14x4 − 3A14x2 + A14 − A7x3 + 2A7x − A6x4 + 3A6x2 − A6 + A3x

Knots in L(9, 2)

S2,∞(42) = −A8x3 + A8x + A4x3 − A4x − x4 + 3x2 − 1 + A−4x2 − A−4

S2,∞(42) = −x3 + 2x + A−4x3 − A−4x − A−8x4 + 3A−8x2 − A−8x − A−8 + A−12x2 − A−12

S2,∞(45) = −A4x3 + 2x3 − x − A−4x4 − A−4x3 + 3A−4x2 − A−4 + A−8x2 + A−8x − A−8

S2,∞(45) = A8x − 2A4x3 + 2A4x + 2x3 − x − A−4x4 + 3A−4x2 − 2A−4x − A−4 + A−8x2 − A−8

S2,∞(414) = A8x −2A4x3+A4x +3x3−2x −A−4x4−A−4x3+3A−4x2−A−4x −A−4+A−8x2+A−8x −A−8

S2,∞(418) = −x3 + x + 2A−4x3 − 2A−4x − A−8x4 − A−8x3 + 3A−8x2 − A−8 + A−12x2 + A−12x − A−12

S2,∞(418) = A12x − 2A8x3 + 2A8x + 2A4x3 − 2A4x − x4 + 3x2 − x − 1 + A−4x2 − A−4

S2,∞(421) = −A4x3 + A4x + x3 − A−4x4 + 3A−4x2 − A−4x − A−4 + A−8x2 − A−8

S2,∞(424) = −A12x3 + 2A12x − A4x4 + A4x3 + 3A4x2 − 2A4x − A4 + x2 − 1S2,∞(55) = A−8x3 − 2A−8x − A−12x4 + 3A−12x2 − A−12x − A−12 + A−16x4 − 2A−16x2 − A−20x2 + A−20

S2,∞(55) = A16x3 − 2A16x − A12x3 + A12x + A8x4 − 3A8x2 + A8 − A4x4 + A4x3 + 2A4x2 − 2A4x + x2 − 1S2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 + A2x4 − 3A2x2 + A2 − A−2x4 + A−2x3 + 2A−2x2 − 2A−2x +A−6x2 − A−6x − A−6

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 + A−10x4 − 2A−10x2 − A−14x4 + A−14x3 + 2A−14x2 −2A−14x + A−18x2 − A−18x − A−18

S2,∞(58) = −x3 + x + 3A−4x3 − 4A−4x − A−8x4 − A−8x3 + 3A−8x2 − A−8x − A−8 + A−12x4 − 2A−12x2 +A−12x − A−16x2 + A−16

S2,∞(58) = A12x3−A12x −2A8x3+A8x +A4x4+2A4x3−3A4x2−2A4x +A4− x4+2x2− x +A−4x2−A−4

S2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 + 2A−2x4 − 4A−2x2 + A−2 − A−6x4 + A−6x3 + A−6x2 −2A−6x + A−6 + A−10x2 − A−10x − A−10

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 + A−6x4 − A−6x2 − A−10x4 +


A−10x3 + 2A−10x2 − 2A−10x + A−14x2 − A−14x − A−14

S2,∞(512) = −x3+4A−4x3−4A−4x−A−8x4−3A−8x3+3A−8x2+A−8x−A−8+A−12x4+A−12x3−2A−12x2+A−12x − A−16x2 − A−16x + A−16

S2,∞(512) = −A16x + 2A12x3 − A12x − 4A8x3 + 3A8x + A4x4 + 3A4x3 − 3A4x2 − 2A4x + A4 − x4 + 2x2 −2x + A−4x2 − A−4

S2,∞(514) = −A4x3 + 3x3 − 2x − A−4x4 − 2A−4x3 + 3A−4x2 − A−4 + A−8x4 + A−8x3 − 2A−8x2 − A−12x2 −A−12x + A−12

S2,∞(514) = −A12x+2A8x3−2A8x−3A4x3+2A4x+x4+2x3−3x2+1−A−4x4+2A−4x2−2A−4x+A−8x2−A−8

S2,∞(515) = −A10x2+2A6x4−2A6x2+A6−3A2x4+3A2x2+2A−2x4−2A−2x2−A−6x4+A−6x3+A−6x2−2A−6x + A−10x2 − A−10x − A−10

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 + 2A−6x4 − 2A−6x2 + A−6 − A−10x4 + A−10x3 +A−10x2 − 2A−10x + A−14x2 − A−14x − A−14

S2,∞(516) = A8x3−3A4x3+2A4x+x4+3x3−3x2−x+1−A−4x4−A−4x3+2A−4x2−A−4x+A−8x2+A−8x−A−8

S2,∞(516) = A8x −2A4x3+A4x +4x3−3x −A−4x4−2A−4x3+3A−4x2−A−4+A−8x4−2A−8x2+2A−8x −A−12x2 + A−12

S2,∞(519) = −A4x3+3x3−3x−A−4x4−A−4x3+3A−4x2−A−4x−A−4+A−8x4−2A−8x2+A−8x−A−12x2+A−12

S2,∞(519) = A8x3 −A8x − 2A4x3 +A4x + x4 + 2x3 − 3x2 − x + 1−A−4x4 + 2A−4x2 − 2A−4x +A−8x2 −A−8

S2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 + 2A−6x4 − 3A−6x2 + A−6 − A−10x4 + A−10x3 + A−10x2 −2A−10x + A−14x2 − A−14x − A−14

S2,∞(520) = −A10x2+2A6x4−3A6x2+A6−2A2x4+2A2x2+A−2x4−A−2x2−A−6x4+A−6x3+2A−6x2−2A−6x + A−10x2 − A−10x − A−10

S2,∞(521) = A12x3 − 3A8x3 + 3A8x + A4x4 + 2A4x3 − 3A4x2 − A4x + A4 − x4 + 2x2 − x + A−4x2 − A−4

S2,∞(521) = −x3 + x + 3A−4x3 − 3A−4x − A−8x4 − 2A−8x3 + 3A−8x2 + A−8x − A−8 + A−12x4 − 2A−12x2 +2A−12x − A−16x2 + A−16

S2,∞(544) = −x3 + x + 3A−4x3 − 3A−4x −A−8x4 − 2A−8x3 + 3A−8x2 −A−8 +A−12x4 +A−12x3 − 2A−12x2 −A−16x2 − A−16x + A−16

S2,∞(544) = −A16x + 2A12x3 − 2A12x − 3A8x3 + 2A8x + A4x4 + 2A4x3 − 3A4x2 − A4x + A4 − x4 + 2x2 −x + A−4x2 − A−4

S2,∞(546) = −x3 + x + 3A−4x3 − 2A−4x − A−8x4 − 3A−8x3 + 3A−8x2 + 2A−8x − A−8 + A−12x4 + A−12x3 −2A−12x2 + A−12x − A−16x2 − A−16x + A−16

S2,∞(546) = −A16x+2A12x3−A12x−4A8x3+4A8x+A4x4+2A4x3−3A4x2+A4−x4+2x2−x+A−4x2−A−4

S2,∞(547) = −A10x2 + 2A6x4 − 2A6x2 − 3A2x4 + 4A2x2 −A2 + 2A−2x4 − 3A−2x2 +A−2 −A−6x4 +A−6x3 +A−6x2 − 2A−6x + A−6 + A−10x2 − A−10x − A−10

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 + 2A−6x4 − 2A−6x2 − A−10x4 +A−10x3 + A−10x2 − 2A−10x + A−14x2 − A−14x − A−14

S2,∞(548) = A4x3−A4x −2x3+2x +A−4x4+2A−4x3−3A−4x2−A−4x +A−4−A−8x4−A−8x3+2A−8x2+A−12x2 + A−12x − A−12

S2,∞(548) = A12x−2A8x3+2A8x+3A4x3−3A4x−x4−x3+3x2−1+A−4x4−2A−4x2+A−4x−A−8x2+A−8

S2,∞(562) = A8x − 2A4x3 + 5x3 − 4x − A−4x4 − 3A−4x3 + 3A−4x2 − A−4 + A−8x4 + A−8x3 − 2A−8x2 +A−8x − A−12x2 − A−12x + A−12

S2,∞(562) = −A12x + 2A8x3 −A8x − 4A4x3 + 2A4x + x4 + 4x3 − 3x2 − 2x + 1−A−4x4 −A−4x3 + 2A−4x2 −2A−4x + A−8x2 + A−8x − A−8

S2,∞(564) = A8x − 2A4x3 +A4x + 4x3 − 2x −A−4x4 − 3A−4x3 + 3A−4x2 +A−4x −A−4 +A−8x4 +A−8x3 −2A−8x2 + A−8x − A−12x2 − A−12x + A−12

S2,∞(564) = −A12x +2A8x3−A8x −4A4x3+3A4x +x4+3x3−3x2+ 1−A−4x4−A−4x3+2A−4x2−A−4x +


A−8x2 + A−8x − A−8

S2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 + 3A−6x4 − 2A−6x2 − A−10x4 + A−10x3 −A−10x2 − 2A−10x + A−10 + A−14x2 − A−14xS2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 + 2A−2x4 − A−2x2 − A−6x4 +A−6x3 + A−6x2 − 2A−6x + A−10x2 − A−10x − A−10

S2,∞(568) = −A8x3 + A8x + 2A4x3 − 3A4x − x4 + 3x2 − x − 1 + A−4x4 − 2A−4x2 − A−8x2 + A−8

S2,∞(568) = A4x3 − 2A4x − x3 + x + A−4x4 + A−4x3 − 3A−4x2 − A−4x + A−4 − A−8x4 + 2A−8x2 − A−8x +A−12x2 − A−12

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 − A−6 + 2A−10x4 − 4A−10x2 + A−10 − A−14x4 + A−14x3 +A−14x2 − 2A−14x + A−14 + A−18x2 − A−18x − A−18

S2,∞(569) = −A14x2+A14+2A10x4−4A10x2+A10−2A6x4+4A6x2−A6+A2x4−2A2x2−A−2x4+A−2x3+2A−2x2 − 2A−2x + A−6x2 − A−6x − A−6

S2,∞(570) = A12x3 − A12x − 2A8x3 + 2A8x + A4x4 + A4x3 − 3A4x2 + A4 − x4 + 2x2 + A−4x2 − A−4

S2,∞(570) = −x3 + 2x + 2A−4x3 − 2A−4x − A−8x4 − A−8x3 + 3A−8x2 − A−8 + A−12x4 − 2A−12x2 + A−12x −A−16x2 + A−16

S2,∞(572) = A−8x3 − A−8x − A−12x4 − A−12x3 + 3A−12x2 + A−12x − A−12 + A−16x4 − 2A−16x2 + A−16x −A−20x2 + A−20

S2,∞(572) = A16x3−A16x−2A12x3+3A12x+A8x4−3A8x2+A8x+A8−A4x4+A4x3+2A4x2−2A4x+x2−1S2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 + A−2x4 − 2A−2x2 − A−6x4 + A−6x3 + 2A−6x2 − 2A−6x +A−10x2 − A−10x − A−10

S2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 + A−6x4 − 2A−6x2 + A−6 − A−10x4 + A−10x3 + 2A−10x2 −2A−10x + A−14x2 − A−14x − A−14

S2,∞(583) = −A4x3+A4x+2x3−x−A−4x4−A−4x3+3A−4x2−A−4+A−8x4−2A−8x2+A−8x−A−12x2+A−12

S2,∞(583) = A8x3 −A8x − 2A4x3 + 2A4x + x4 + x3 − 3x2 + x + 1−A−4x4 + 2A−4x2 −A−4x +A−8x2 −A−8

S2,∞(584) = A−2x4 − 2A−2x2 +A−2 − 2A−6x4 + 3A−6x2 + 2A−10x4 − 3A−10x2 −A−14x4 +A−14x3 +A−14x2 −2A−14x + A−18x2 − A−18x − A−18

S2,∞(584) = −A14x2+2A10x4−3A10x2−2A6x4+3A6x2+A2x4−2A2x2+A2−A−2x4+A−2x3+2A−2x2−2A−2x + A−6x2 − A−6x − A−6

S2,∞(585) = A−14x4 − 3A−14x2 + A−14 − A−18x4 + A−18x3 + 2A−18x2 − 2A−18x + A−22x2 − A−22x − A−22

S2,∞(585) = A14x4 − 3A14x2 + A14 − A2x4 + A2x3 + 2A2x2 − 2A2x + A−2x2 − A−2x − A−2

Knots in L(10, 1)

S2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 − A4x4 + 3A4x2 − A4 + x2 − 1S2,∞(56) = A−2x4−3A−2x2+A−2−A−6x4+2A−6x2−A−8x4+3A−8x2−A−8+A−10x2−A−10+A−12x2−A−12

S2,∞(59) = A6x4−A6x2−2A2x4+3A2x2−A2−x4+3x2− 1+A−2x4−A−2x2+A−4x2−A−4−A−6x2+A−6

S2,∞(59) = −A6x2+A6+2A2x4−4A2x2+A2−2A−2x4+3A−2x2−A−2−A−4x4+3A−4x2−A−4+2A−6x2−A−6 + A−8x2 − A−8

S2,∞(515) = −A10x2+2A6x4−2A6x2+A6−3A2x4+3A2x2−x4+3x2− 1+A−2x4+A−2x2−A−2+A−4x2−A−4 − A−6x2

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 − A−4x4 + 3A−4x2 − A−4 + A−6x4 + A−6x2 +A−8x2 − A−8 − A−10x2

S2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 − A−4x4 + 3A−4x2 − A−4 + A−6x4 + A−8x2 − A−8 − A−10x2

180 Knots in L(10, 3)

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 +A6 − 2A2x4 + 2A2x2 − x4 + 3x2 − 1+ 2A−2x2 −A−2 +A−4x2 −A−4

S2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2−x4+3x2−1+A−2x4+A−4x2−A−4−A−6x2+A−6

S2,∞(547) = −A6x2+A6+2A2x4−3A2x2+A2−3A−2x4+4A−2x2−A−2−A−4x4+3A−4x2−A−4+A−6x4+A−6x2 − A−6 + A−8x2 − A−8 − A−10x2

S2,∞(565) = −A6x2+2A2x4−A2x2−4A−2x4+4A−2x2−A−2−A−4x4+3A−4x2−A−4+2A−6x4+A−6x2−A−6 + A−8x2 − A−8 − 3A−10x2 + A−10 + A−14

S2,∞(565) = A14 − 3A10x2 +A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 −A2 − x4 + 3x2 − 1+A−2x4 + 2A−2x2 −A−2 + A−4x2 − A−4 − A−6x2

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 − A−6 − A−8x4 + 3A−8x2 − A−8 + A−10x4 − A−10x2 +A−12x2 − A−12 − A−14x2 + A−14

S2,∞(569) = −A14x2+A14+2A10x4−4A10x2+A10−2A6x4+4A6x2−A6−A4x4+3A4x2−A4+A2x2−A2+x2−1S2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 − x4 + 3x2 − 1 + A−2x2 − A−2 + A−4x2 − A−4

S2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 − A−4x4 + 3A−4x2 − A−4 + A−6x2 + A−8x2 − A−8

S2,∞(584) = A−2x4−2A−2x2+A−2−2A−6x4+3A−6x2−A−8x4+3A−8x2−A−8+A−10x4−A−10+A−12x2−A−12 − A−14x2

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 − A4x4 + 3A4x2 − A4 + A2x2 + x2 − 1S2,∞(585) = A14x4 − 3A14x2 + A14 − A8x4 + 3A8x2 − A8 − A6x4 + 3A6x2 − A6 + A4x2 − A4

Knots in L(10, 3)

S2,∞(56) = A10x4 − 2A10x2 −A6x4 + 2A6x2 +A2x4 − 3A2x2 +A2 − x4 + 3x2 − 1−A−2x4 + 3A−2x2 −A−2 +A−4x2 − A−4 − A−6 + A−10

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 + A−10x4 − 2A−10x2 − A−12x4 + 3A−12x2 − A−12 −A−14x4 + 3A−14x2 − A−14 + A−16x2 − A−16 − A−18 + A−22

S2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 + 2A−2x4 − 4A−2x2 + A−2 − A−4x4 + 3A−4x2 − A−4 −A−6x4 + 2A−6x2 + A−8x2 − A−8 − A−10 + A−14

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 + A−6x4 − A−6x2 − A−8x4 +3A−8x2 − A−8 − A−10x4 + 3A−10x2 − A−10 + A−12x2 − A−12 − A−14 + A−18

S2,∞(515) = −A10x2 +2A6x4 −2A6x2 +A6 − 3A2x4 + 3A2x2 +2A−2x4 −2A−2x2 −A−4x4 + 3A−4x2 −A−4 −A−6x4 + 2A−6x2 − A−6 + A−8x2 − A−8 − A−10 + A−14

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 + 2A−6x4 − 2A−6x2 + A−6 − A−8x4 + 3A−8x2 −A−8 − A−10x4 + 2A−10x2 − A−10 + A−12x2 − A−12 − A−14 + A−18

S2,∞(520) = A2x4−A2x2−2A−2x4+2A−2x2+2A−6x4−3A−6x2+A−6−A−8x4+3A−8x2−A−8−A−10x4+2A−10x2 − A−10 + A−12x2 − A−12 − A−14 + A−18

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 + A6 − 2A2x4 + 2A2x2 + A−2x4 − A−2x2 − A−4x4 + 3A−4x2 − A−4 −A−6x4 + 3A−6x2 − A−6 + A−8x2 − A−8 − A−10 + A−14

S2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2+2A−2x4−3A−2x2+A−2−A−4x4+3A−4x2−A−4 − A−6x4 + 2A−6x2 + A−8x2 − A−8 − A−10 + A−14

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 + 2A−6x4 − 2A−6x2 − A−8x4 +3A−8x2 − A−8 − A−10x4 + 2A−10x2 − A−10 + A−12x2 − A−12 − A−14 + A−18

S2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 + 3A−6x4 − 2A−6x2 − A−8x4 + 3A−8x2 −A−8 − A−10x4 + A−12x2 − A−12 + A−18

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 + 2A−2x4 − A−2x2 − A−4x4 +


3A−4x2 − A−4 − A−6x4 + 2A−6x2 − A−6 + A−8x2 − A−8 − A−10 + A−14

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 −A−6 + 2A−10x4 − 4A−10x2 +A−10 −A−12x4 + 3A−12x2 −A−12 − A−14x4 + 2A−14x2 + A−16x2 − A−16 − A−18 + A−22

S2,∞(569) = −A14x2 + A14 + 2A10x4 − 4A10x2 + A10 − 2A6x4 + 4A6x2 − A6 + A2x4 − 2A2x2 − x4 + 3x2 −1 − A−2x4 + 3A−2x2 − A−2 + A−4x2 − A−4 − A−6 + A−10

S2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 + A−2x4 − 2A−2x2 − A−4x4 + 3A−4x2 − A−4 − A−6x4 +3A−6x2 − A−6 + A−8x2 − A−8 − A−10 + A−14

S2,∞(573) = A2x4 − 2A2x2 −A−2x4 +A−2x2 +A−6x4 − 2A−6x2 +A−6 −A−8x4 + 3A−8x2 −A−8 −A−10x4 +3A−10x2 − A−10 + A−12x2 − A−12 − A−14 + A−18

S2,∞(584) = A−2x4 − 2A−2x2 +A−2 − 2A−6x4 + 3A−6x2 + 2A−10x4 − 3A−10x2 −A−12x4 + 3A−12x2 −A−12 −A−14x4 + 2A−14x2 − A−14 + A−16x2 − A−16 − A−18 + A−22

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 + A2x4 − 2A2x2 + A2 − x4 + 3x2 − 1 − A−2x4 +3A−2x2 − A−2 + A−4x2 − A−4 − A−6 + A−10

S2,∞(585) = A−14x4 − 3A−14x2 + A−14 − A−16x4 + 3A−16x2 − A−16 − A−18x4 + 3A−18x2 − A−18 + A−20x2 −A−20 − A−22 + A−26

S2,∞(585) = A14x4 − 3A14x2 + A14 − A4x4 + 3A4x2 − A4 − A2x4 + 3A2x2 − A2 + x2 − 1 − A−2 + A−6

Knots in L(11, 1)

S2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 − A5x5 + 4A5x3 − 3A5x + Ax3 − 2AxS2,∞(56) = A−2x4−3A−2x2+A−2−A−6x4+2A−6x2−A−7x5+4A−7x3−3A−7x+A−10x2−A−10+A−11x3−2A−11xS2,∞(59) = A6x4 −A6x2 − 2A2x4 + 3A2x2 −A2 −Ax5 + 4Ax3 − 3Ax +A−2x4 −A−2x2 +A−3x3 − 2A−3x −A−6x2 + A−6

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 − A−3x5 + 4A−3x3 − 3A−3x +2A−6x2 − A−6 + A−7x3 − 2A−7xS2,∞(515) = −A10x2 + 2A6x4 − 2A6x2 +A6 − 3A2x4 + 3A2x2 −Ax5 +4Ax3 − 3Ax +A−2x4 +A−2x2 −A−2 +A−3x3 − 2A−3x − A−6x2

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 − A−3x5 + 4A−3x3 − 3A−3x + A−6x4 + A−6x2 +A−7x3 − 2A−7x − A−10x2

S2,∞(520) = A2x4−A2x2−2A−2x4+2A−2x2−A−3x5+4A−3x3−3A−3x+A−6x4+A−7x3−2A−7x−A−10x2

S2,∞(520) = −A10x2+2A6x4−3A6x2+A6−2A2x4+2A2x2−Ax5+4Ax3−3Ax+2A−2x2−A−2+A−3x3−2A−3xS2,∞(547) = −A10x2 + 2A6x4 − 2A6x2 − 3A2x4 + 4A2x2 − A2 − Ax5 + 4Ax3 − 3Ax + A−2x4 + A−3x3 −2A−3x − A−6x2 + A−6

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 − A−3x5 + 4A−3x3 − 3A−3x +A−6x4 + A−6x2 − A−6 + A−7x3 − 2A−7x − A−10x2

S2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 − A−3x5 + 4A−3x3 − 3A−3x + 2A−6x4 +A−6x2 − A−6 + A−7x3 − 2A−7x − 3A−10x2 + A−10 + A−14

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 − Ax5 + 4Ax3 − 3Ax + A−2x4 +2A−2x2 − A−2 + A−3x3 − 2A−3x − A−6x2

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 − A−6 − A−7x5 + 4A−7x3 − 3A−7x + A−10x4 − A−10x2 +A−11x3 − 2A−11x − A−14x2 + A−14

S2,∞(569) = −A14x2 +A14 + 2A10x4 −4A10x2 +A10 − 2A6x4 +4A6x2 −A6 −A5x5 +4A5x3 − 3A5x +A2x2 −

182 Knots in L(11, 2)

A2 + Ax3 − 2AxS2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 − Ax5 + 4Ax3 − 3Ax + A−2x2 − A−2 + A−3x3 − 2A−3xS2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 − A−3x5 + 4A−3x3 − 3A−3x + A−6x2 + A−7x3 − 2A−7xS2,∞(584) = A−2x4 − 2A−2x2 + A−2 − 2A−6x4 + 3A−6x2 − A−7x5 + 4A−7x3 − 3A−7x + A−10x4 − A−10 +A−11x3 − 2A−11x − A−14x2

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 − A5x5 + 4A5x3 − 3A5x + A2x2 + Ax3 − 2AxS2,∞(585) = A14x4 − 3A14x2 + A14 − A9x5 + 4A9x3 − 3A9x − A6x4 + 3A6x2 − A6 + A5x3 − 2A5x

Knots in L(11, 2)

S2,∞(56) = A10x4 − 2A10x2 − A6x4 + 2A6x2 + A2x5 − 4A2x3 + 3A2x − A−2x3 + 2A−2xS2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 + A−10x5 − 4A−10x3 + A−10x2 + 3A−10x − A−10 −A−14x3 + 2A−14xS2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 + A−2x5 + A−2x4 − 4A−2x3 − A−2x2 + 3A−2x − A−6x3 −A−6x2 + 2A−6x + A−6

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 + A−6x5 − 4A−6x3 + 2A−6x2 +3A−6x − A−6 − A−10x3 + 2A−10xS2,∞(515) = −A10x2 +2A6x4 −2A6x2 +A6 − 3A2x4 + 3A2x2 +A−2x5 +A−2x4 −4A−2x3 +A−2x2 + 3A−2x −A−2 − A−6x3 − A−6x2 + 2A−6xS2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 + A−6x5 + A−6x4 − 4A−6x3 + A−6x2 + 3A−6x −A−10x3 − A−10x2 + 2A−10xS2,∞(520) = A2x4−A2x2−2A−2x4+2A−2x2+A−6x5+A−6x4−4A−6x3+3A−6x−A−10x3−A−10x2+2A−10xS2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 +A6 − 2A2x4 + 2A2x2 +A−2x5 − 4A−2x3 + 2A−2x2 + 3A−2x −A−2 −A−6x3 + 2A−6xS2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2+A−2x5+A−2x4−4A−2x3+3A−2x −A−6x3−A−6x2 + 2A−6x + A−6

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 + A−6x5 + A−6x4 − 4A−6x3 +A−6x2 + 3A−6x − A−6 − A−10x3 − A−10x2 + 2A−10xS2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 + A−6x5 + 2A−6x4 − 4A−6x3 + A−6x2 +3A−6x − A−6 − A−10x3 − 3A−10x2 + 2A−10x + A−10 + A−14

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 + A−2x5 + A−2x4 − 4A−2x3 +2A−2x2 + 3A−2x − A−2 − A−6x3 − A−6x2 + 2A−6xS2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 + 4A−6x2 −A−6 +A−10x5 +A−10x4 − 4A−10x3 −A−10x2 + 3A−10x −A−14x3 − A−14x2 + 2A−14x + A−14

S2,∞(569) = −A14x2 +A14 + 2A10x4 −4A10x2 +A10 − 2A6x4 +4A6x2 −A6 +A2x5 −4A2x3 +A2x2 + 3A2x −A2 − A−2x3 + 2A−2xS2,∞(573) = A6x4 −2A6x2 +A6 −A2x4 +A2x2 +A−2x5 −4A−2x3 +A−2x2 + 3A−2x −A−2 −A−6x3 +2A−6xS2,∞(573) = A2x4 − 2A2x2 − A−2x4 + A−2x2 + A−6x5 − 4A−6x3 + A−6x2 + 3A−6x − A−10x3 + 2A−10xS2,∞(584) = A−2x4 − 2A−2x2 + A−2 − 2A−6x4 + 3A−6x2 + A−10x5 + A−10x4 − 4A−10x3 + 3A−10x − A−10 −A−14x3 − A−14x2 + 2A−14xS2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 +A2x5 − 4A2x3 +A2x2 + 3A2x −A−2x3 + 2A−2xS2,∞(585) = A14x4 − 3A14x2 + A14 + A6x5 − A6x4 − 4A6x3 + 3A6x2 + 3A6x − A6 − A2x3 + 2A2x


Knots in L(11, 3)

S2,∞(56) = A10x4−2A10x2−A6x4+2A6x2−A3x5+4A3x3−3A3x+A2x4−3A2x2+A2−A−2x4+3A−2x2−A−2 + A−5x3 − A−5x − A−6 − A−9x + A−10

S2,∞(56) = A−2x4 − 3A−2x2 + A−2 − A−6x4 + 2A−6x2 − A−9x5 + 4A−9x3 − 3A−9x + A−10x4 − 2A−10x2 −A−14x4 + 3A−14x2 − A−14 + A−17x3 − A−17x − A−18 − A−21x + A−22

S2,∞(59) = A6x4 − A6x2 − 2A2x4 + 3A2x2 − A2 − A−1x5 + 4A−1x3 − 3A−1x + 2A−2x4 − 4A−2x2 + A−2 −A−6x4 + 2A−6x2 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(59) = −A6x2 + A6 + 2A2x4 − 4A2x2 + A2 − 2A−2x4 + 3A−2x2 − A−2 − A−5x5 + 4A−5x3 − 3A−5x +A−6x4 − A−6x2 − A−10x4 + 3A−10x2 − A−10 + A−13x3 − A−13x − A−14 − A−17x + A−18

S2,∞(515) = −A10x2+2A6x4−2A6x2+A6−3A2x4+3A2x2−A−1x5+4A−1x3−3A−1x+2A−2x4−2A−2x2−A−6x4 + 2A−6x2 − A−6 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(515) = −A6x2 + 2A2x4 − 2A2x2 − 3A−2x4 + 3A−2x2 −A−5x5 + 4A−5x3 − 3A−5x + 2A−6x4 − 2A−6x2 +A−6 − A−10x4 + 2A−10x2 − A−10 + A−13x3 − A−13x − A−14 − A−17x + A−18

S2,∞(520) = A2x4 − A2x2 − 2A−2x4 + 2A−2x2 − A−5x5 + 4A−5x3 − 3A−5x + 2A−6x4 − 3A−6x2 + A−6 −A−10x4 + 2A−10x2 − A−10 + A−13x3 − A−13x − A−14 − A−17x + A−18

S2,∞(520) = −A10x2 + 2A6x4 − 3A6x2 +A6 − 2A2x4 + 2A2x2 −A−1x5 +4A−1x3 − 3A−1x +A−2x4 −A−2x2 −A−6x4 + 3A−6x2 − A−6 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(547) = −A10x2+2A6x4−2A6x2−3A2x4+4A2x2−A2−A−1x5+4A−1x3−3A−1x+2A−2x4−3A−2x2+A−2 − A−6x4 + 2A−6x2 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(547) = −A6x2 + A6 + 2A2x4 − 3A2x2 + A2 − 3A−2x4 + 4A−2x2 − A−2 − A−5x5 + 4A−5x3 − 3A−5x +2A−6x4 − 2A−6x2 − A−10x4 + 2A−10x2 − A−10 + A−13x3 − A−13x − A−14 − A−17x + A−18

S2,∞(565) = −A6x2 + 2A2x4 − A2x2 − 4A−2x4 + 4A−2x2 − A−2 − A−5x5 + 4A−5x3 − 3A−5x + 3A−6x4 −2A−6x2 − A−10x4 + A−13x3 − A−13x − A−17x + A−18

S2,∞(565) = A14 − 3A10x2 + A10 + 3A6x4 − 2A6x2 − 4A2x4 + 4A2x2 − A2 − A−1x5 + 4A−1x3 − 3A−1x +2A−2x4 − A−2x2 − A−6x4 + 2A−6x2 − A−6 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(569) = A−2x4 − 2A−2x2 − 2A−6x4 +4A−6x2 −A−6 −A−9x5 +4A−9x3 − 3A−9x + 2A−10x4 −4A−10x2 +A−10 − A−14x4 + 2A−14x2 + A−17x3 − A−17x − A−18 − A−21x + A−22

S2,∞(569) = −A14x2 +A14 + 2A10x4 −4A10x2 +A10 − 2A6x4 +4A6x2 −A6 −A3x5 +4A3x3 − 3A3x +A2x4 −2A2x2 − A−2x4 + 3A−2x2 − A−2 + A−5x3 − A−5x − A−6 − A−9x + A−10

S2,∞(573) = A6x4 − 2A6x2 + A6 − A2x4 + A2x2 − A−1x5 + 4A−1x3 − 3A−1x + A−2x4 − 2A−2x2 − A−6x4 +3A−6x2 − A−6 + A−9x3 − A−9x − A−10 − A−13x + A−14

S2,∞(573) = A2x4−2A2x2−A−2x4+A−2x2−A−5x5+4A−5x3−3A−5x+A−6x4−2A−6x2+A−6−A−10x4+3A−10x2 − A−10 + A−13x3 − A−13x − A−14 − A−17x + A−18

S2,∞(584) = A−2x4 − 2A−2x2 +A−2 − 2A−6x4 + 3A−6x2 −A−9x5 + 4A−9x3 − 3A−9x + 2A−10x4 − 3A−10x2 −A−14x4 + 2A−14x2 − A−14 + A−17x3 − A−17x − A−18 − A−21x + A−22

S2,∞(584) = −A14x2 + 2A10x4 − 3A10x2 − 2A6x4 + 3A6x2 − A3x5 + 4A3x3 − 3A3x + A2x4 − 2A2x2 + A2 −A−2x4 + 3A−2x2 − A−2 + A−5x3 − A−5x − A−6 − A−9x + A−10

S2,∞(585) = −A−13x5 +4A−13x3 − 3A−13x +A−14x4 − 3A−14x2 +A−14 −A−18x4 + 3A−18x2 −A−18 +A−21x3 −A−21x − A−22 − A−25x + A−26

S2,∞(585) = A14x4−3A14x2+A14−A7x5+4A7x3−3A7x−A2x4+3A2x2−A2+A−1x3−A−1x−A−2−A−5x+A−6

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Index

(p, q)-curve, 30−K, 161, 51

H, 42J, 19X, 21, 51#, 16

∆, 19, 53

I , 53X , 19K, 17Bp,q, 47

Bp, 42

C, 25Hi , 55

L, 22, 23S , 22χ, 26, 55S3, 24, 39, 197S2,∞, 23, 49⟨ q⟩, 19J qK, 26, 52L(p, q), 29Sfr, 23∇, 19P , 27, 50SL, 33, 36

SLn, 44q[s] , 25q{l}, 25q{l ,m}, 51∣s∣, 19∧, 50wind, 72wr, 16cr, 16dξ, 52

m, 53n±, 19

algorithm

classi�cation, 66, 69

optimization, 67

arrow, 34

bad, 42

cancellation, 35

diagram, 34

good, 42

pushing, 35

bifurcation, 52

binary search tree, 68

canonical form, 67

categori�cation

Jones polynomial, 25

KBSM, 49

virual links, 49

chain

complex, 25, 50

group, 25, 51

circle, 19

projective, 50

classi�cation, 18, 70

knots in L(p, q), 69knots in solid torus, 66

connected sum, 16

crossing, 14

number, 16

sign, 16

smoothening of, 19

cube, 52

discrete, 19, 26

edge of, 51

degree shi�, 25

diagram

arrow, 34

disk, 31

knot, 14

punctured disk, 33

189

190 Index

standard, 43

di�erential, 26, 51

partial, 52

total, 54

embedding, 13

ambient isotopic, 14

�attening

of a cube, 61

of a module, 51

of a partial di�erential, 54

�ip, 58

�ype, 15, 66

framing

blackboard, 17

number, 17

fundamental group, 71

Gauss code, 64

divisible, 68

extended, 64

realizable, 64

reduction, 65

Gauss word, 63

graded

dimension, 25

Euler characteristic, 26, 55

vector space, 25

height shi�, 25

homology

group, 55

Khovanov, 25, 50, 55

isotopy

ambient, 14

regular, 17

Kau�man

bracket, 19

relation, 20

triple, 20

Khovanov

chain complex, 25

homology, 25, 50

kink, 15

knot, 13

a�ne, 17

amphichiral, 17, 69

chiral, 17

composition, 17

diagram, 14

equivalent, 15

framed, 17

invariant, 14

mirror, 17

notation, 63

oriented, 15

polynomial, 18

prime, 17

reverse, 16

table, 18, 70

trivial, 15

wild, 14

lens space, 29

homeomorphic, 31

homotopic, 31

lexicographical ordering, 64

link, 13

empty, 24

invertible, 71

table, 18

locally consistent orientation, 52

Möbius band, 52

module, see skein module

operator

comultiplication, 53

degree shi�, 25, 51

height shi�, 25

multiplication, 53

oval, 39

partition, 66

permutation

cyclic, 58

rule, 50

polynomial

X, 21Alexander, 18

Alexander-Conway, 19

Index 191

bracket, 19

HOMFLYPT, 21

Jones, 19

normalized Alexander, 19

Poincaré, 27, 50

unnormalized Jones, 21

projection

regular, 14

regular projection, 14

Reidemeister moves, 15, 65

arrow diagram, 36

disk diagram, 32

punctured disk diagram, 33

relation

framing, 23

HOMFLYPT, 24

Kau�man, 20, 23

skein, 19

ribbon, 17

semi identity, 53

shadow, 14

skein

relation, 19

triple, 19

skein module, 22

HOMFLYPT, 24, 39

Kau�man bracket, 23, 49

slide, 33, 36

standard, 43

solid torus, 30

space

lens, 29

projective, 30

state, 19

state-sum formula, 19

symmetric graph, 58

unknot, 15

wedge product, 50

winding number, 72

writhe, 16

classification od knots in lens spaces

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