New directions in Combinatorial Knot Theory

ROGER FENN1

1School of Mathematical Sciences, University of Sussex

Falmer, Brighton, BN1 9RH, England

e-mail address: rogerf@sussex.ac.uk

ABSTRACT

Combinatorial Knot Theory:The study of knotsby the use of diagrams.

Diagrams, since the time of Gauss and earlierhave always been used to represent knots. Thisis because a diagram of a knot on a 2 dimen-sional plane gives an immediate and physical feelfor the knot in 3 dimensional space.

The Anglo-Saxon and Celtic people were trulyinventive at using planar pictorial means to rep-resent the 3 dimensional intertwinings of cordsand thongs as were other cultures.

Many topological invariants of knots such as thefundamental group, the Alexander polynomial,the Jones’ polynomial etc can be investigatedfrom a diagram of the knot in question. Otherinvariants, such as the unknotting number, thecrookedness, the energy, etc struggle to be de-fined, if at all, from the diagram.

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Recently, diagrams have evolved to represent ob-jects other than classical knots. Indeed some di-agrams are objects in their own right and repre-sent no physical object. These evolved diagramshave spawned a huge interest in previously un-defined algebras of a combinatorial nature.

In this talk I shall put these diagrams into amathematical context. There is too little timeto consider all their associated algebras. But Ishall look at the fundamental birack and biquan-dle which appear to be purely combinatorial andwithout topological interpretation.

2

Search in ArXiv for papers with virtual andknot in the title.

None before 1998

from 1998 to 2011 the mean per annum was lessthan 4

in 2012 there were 55

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Knot PropertiesTOPOLOGICAL MIXED COMBINATORIAL

crookedness group crossing numbertotal curvature quandle biquandle

unknotting number genusstick number Alexander

volume Jonestunnel number Homfly(?)

4

Here is a trefoil knot and its mirror image asthe ancients would have represented them. Thecontinuous arcs broken to represent their pas-sage under other arcs.

So classically a knot diagram is the projectedimage in the plane of a simple closed curve inspace; the image having double points where 2distinct points coalesce into 1 point. The brokennature of the arc at a double point tells us whichpoint is above the other in space.

We can now try and make this concept mathe-matical and generalise at the same time.

5

A knot diagram is an oriented 4-valent graphon a surface with crossing type decorations ateach vertex.

Positive and Negative Classical Crossings anda Virtual or Weld Crossing

Positive and Negative Singular Crossings anda Flat Crossing

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Any (non-oriented) 4-valent graph is the imageof immersed circle(s), and so has implied orien-tation of the edges so that each crossing orien-tation is axial.

Axial Orientation

Let us call the orientation of a diagram with allcrossings axially oriented, natural.

However there are 5 more possibilities.

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Source, Sink and Saddle

3 In and 3 Out

8

Classical knot theory can be defined by

Knot=Diagram/Reidemeister Moves

So a Reidemeister move changes one diagraminto another. Two diagrams represent the sameknot if there is a chain of Reidemeister moveschanging one diagram into the other.

Let us make this notion clear and generalise atthe same time.

9

A groupoid with moves has all its morphismsgenerated by a collection of moves. Some of themoves, called null, form a subgroupoid. Themoves are irreducible in the sense that if a moveis a product of two morphisms then one of themis null.

A groupoid, G , with moves is said to have levelsif each object K ∈ G has a level, |K| which isan element of a totally ordered set.

Let m : K −→ L be a move. Then if |K| < |L|we call m an expansion and write K

m

ր L .If |K| > |L| we call m a collapse and write

Km

ց L . Note that the inverse of an expansionis a collapse. If |K| = |L| then we call the movem level preserving. All null moves are levelpreserving.

If K and L are in the same component thenthey are called equivalent and we write K ≃ L .If K and L are related by a sequence of levelpreserving moves then they are called isomor-phic and we write K ∼= L . So an isomorphismis a finer kind of equivalence.

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The diamond property says that given a se-quence of morphisms K ր K ′ ∼= L′ ց L theneither K ∼= L or there is a sequence K ց K ′′ ∼=L′′ ր L

K ′ ∼= L′

ր ցK L

ց րK ′′ ∼= L′′

An object K is minimal if there is no sequenceK ∼= K ′ ց L . The following is now easy toprove.

Let G be a groupoid with the diamond prop-erty. If K and L are equivalent minimal ob-jects then they are isomorphic and |K| = |L| .Alternatively, any equivalence class has a min-imal element which is unique up to isomor-phism.

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The idea of a groupoid with the diamond prop-erty has many applications. To get a glimpse ofits universality here are two examples.

Free Groups The objects are words in the sym-bols x ∈ X and x−1 ∈ X−1 . The level of aword is its length. The expansions are inser-tions of pairs xx−1 or x−1x in the words. Theisomorphisms are equality. Then every word isequivalent to a unique reduced word.

The singular braid monoid embeds in a group.(Rimanyi, Rourke, F )

Here the objects are singular braids. The nullmoves are the braid generators and the expan-sions are the introduction of pairs of cancellingsingular crossings. The levels are the number ofsingular crossings.

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The moves on a diagram can be grouped into 5classes. The null moves form one class.

1. Birth or Death of a Curl

2. Inverse Vertices

3. Dominating Vertices

4. Commuting Vertices

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In the next example the objects are diagrams inthe plane with flat crossings. So a diagram rep-resents a doodle. The moves are of type 1 and2 but not 3, (no triple points are allowed). Thenull moves are homeomorphisms of the planetaking one diagram to another and if a small cir-cle exists as a component then it can be movedanywhere in the plane to another small circle.(Small means that it has no part of the diagraminside it.)

The level of a doodle is the number of crossings.

The Poppy, level 8

The Borromean Doodle, level 6

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Unfortunately this doesn’t work for classical knotdiagrams with level equal to the number of cross-ings.

Unknots

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However there is something we can say aboutvirtual knots. We first need a geometric inter-pretation of virtual diagrams due to S. and N.Kamada.

Embed the edges in ribbons and at a classicalcrossing do the following.

At a virtual crossing do this.

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For example, this virtual diagram

embeds in this surface.

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So we have eliminated the virtual crossings andwe have a “classical’’ knot diagram K on a sur-face Σ . By capping off the boundary compo-nents we can assume that Σ is closed.

Let |K| be the genus of Σ . The null movesare Reidemeister moves on K and homeomor-phisms of Σ taking one diagram to another.The other moves which raise or lower |K| con-sist of attaching or eliminating handles on Σwhich are disjoint from K . The resulting groupoidhas all the properties of a groupoid with movesand we therefore have the following theorem.

If two virtual knots on surfaces of minimalgenus are equivalent then they are equivalentby moves which do not raise the genus.(Kuperberg)

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So far we have decorated the vertices of the di-agram with various kinds of crossings. We nowdecorate the edges with labels or colours andsee how they behave under the different movesallowed.

Suppose we have a crossing • with a naturalorientation and the incoming edges are labelleda, b and the outgoing edges are labelled c, d withelements of some labelling set X .

b

a c

d

The outgoing labels are given by a function S•

defined on pairs of elements of X by S•(a, b) =(c, d) . The function S• is called the switch as-sociated to the crossing • .

We can also define a function F• by F•(b, d) =(a, c) . F is called the sideways map.

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The classes now correspond to the following re-lations

1.Suppose S•(a, b) = (a, d) then b = d

and conversely.

Alternatively, F•(∆) = ∆ . Where ∆ is the di-agonal.

2.S•S• = S•S• = id

F•F• = F•F• = id

3.

(S• × id)(id× S•)(S• × id) = (id× S•)(S• × id)(id× S•)

4.S•S• = S•S•

F•F• = F•F•

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We can also interpret conditions 1. to 3. asfollows. Consider the binary operations definedby F (a, b) = (ba, a

b) . So S(xy, y) = (yx, x) . Then(a, b) −→ ab and (b, a) −→ ba , are called thedown and up operations.

1. aa = aa

2. The down and up operations are right invert-ible.

3. i) abcb = acbc

, ii) cbab = caba and iii) baca = bcac

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The operations have a pleasing symmetry.

If we write the operations as a∧b and a∨b thenwe have

1. a ∧ a = a ∨ a

3. b ∧ (c ∨ b) acts up like c ∧ (b ∧ c) acts up

b ∨ (a ∧ b) acts down like a ∨ (b ∨ a) acts down

a ∧ (c ∨ a) acts down like c ∨ (a ∧ c) acts up

Two binary operations satisfying 1. to 3. abovedefine a biquandle. If the condition 1. is notnecessarily satisfied it is called a birack.

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Let us see how biquandle labelling of the edgesapplies to classical crossings under the Reide-meister moves.

aba

ab

bba

−aba

ab

b ba

a

b a

b

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aaa

aa

aaa

i)

a

b

−yb

y

+yd

x

c

d

ii)

a

b

+by

y

−dy

x

c

d

iii)

a

b

+yx

y

−yx

x

c

d

iv)

a

b

+aby

−cd

x

c

d

a

c

bc

abcb

cbab

ba

ab

cb

b

a

c

bc

acb

c

caba

baca

baca

ac

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If we let the switch for a virtual crossing be theinterchange

S =

(

0 11 0

)

Then possibilities for the classical crossings are

S =

(

1− st ts 0

)

Generalised Alexander Polynomials

S =

(

1 + i tj−t−1j 1 + i

)

Budapest Polynomials

ab = ab = a+ 1, a, b ∈ Z

Cheng labelling

bb = bw = bb = bw = w, wb = ww = ww = wb = b

Black and White labelling

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For doodles, there is a sequence of 2-variablepolynomials defined by

S =

(√1− st st −

√1− st

)

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Biquandle Homology

Let X be the labels of a birack. A word w =a1 · · · an of length n is called an n -cube. Forexample a 1-cube is a label, a 2-cube is a crossingand a 3-cube is a Reidemeister III move.

For each n -cube there are 2n faces of dimensionn− 1 ,

∂−

i (a1 · · · an) = a1 · · · ai−1ai+1 · · · an and

∂+i (a1 · · · an) = (a1)

ai · · · (ai−1)ai(ai+1)ai

· · · (an)ai

for i = 1, . . . , n .

A cubical cell complex, ΓX , can be defined inthe usual way by identifying all the disjoint cubeshaving common faces. Properties of this com-plex can now be used to define knot invariants.For example the fundamental group has a pre-sentation

π1(ΓX) = 〈x, y ∈ X | xyx = yxy〉.

Question If X is the free rack, is ΓX a cat(0)space? A positive answer would imply that allits higher homotopy groups are trivial.

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Let Cn = Cn(ΓX) be the free abelian group withbasis the n -cubes of ΓX . The homomorphism∂ : Cn −→ Cn is defined on cubes by

∂ =n∑

i=1

(−1)i(∂−

i − ∂+i )

and extended linearly. The Yang-Baxter equa-tions then imply that the composition

Cn −→ Cn−1 −→ Cn−2

is zero, and so we have a chain complex. Thehomology of this chain complex H∗(C) and thehomology of any subchain complex is thereforean invariant.

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One important example is the degenerate sub-chain. A cube, a1 · · · an , is called degenerate ifai = ai+1 , for some i = 1, . . . , n−1 . If the birackis a biquandle, so aa = aa for all a , then thedegenerate cubes generate a subchain, D∗ . Thequotient BQ∗ = C∗/D∗ defines the biquandlechain complex and the short exact sequence

0 −→ D∗ −→ C∗ −→ BQ∗ −→ 0

extends to a long exact homology sequence.

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Another example is the double of a birack. Con-sider the set of pairs of pairs X2 × X2 . ThenW ⊂ X2 ×X2 is the set of pairs of pairs definedby

W = {ac, bc | a, b, c,∈ X, ca = cb}.The doubled operations are

(ac)(bc) = abcb, (bc)(ac) = baca

andG(ac, bc) = (bac

a, abca)

Doubling converts racks into biracks and quan-dles into biquandles.

The homology of the double of the 3-colour quan-dle can be used to distinguish the right and lefttrefoils.

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Much work has been done on the homology ofracks and quandles. However the homology ofbiracks and biquandles is little understood andneeds investigating.

Let D be a knot diagram which has its edgescoloured by a biquandle X . Each crossing isassociated to a pair ab ∈ X2 . The sum withsigns

ζ =∑

±ab

is a 2-cycle in the classifying space.

Its homology class is a knot invariant. For exam-ple if we double the 3-colour quandle and applythe resulting biquandle to the two trefoils thecycles are

br

rb gg

brgr

rr

br

gg rb

grbr

rr

(br)(gr)+(rr)(br)+(gr)(rr) and −(br)(gr)−(rr)(br)−(gr)(rr)

which define distinct classes.

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