Adv. Geom. 4 (2004), 263–277 Advances in Geometry( de Gruyter 2004

(1, 1)-knots via the mapping class group ofthe twice punctured torus

Alessia Cattabriga and Michele Mulazzani

(Communicated by G. Gentili)

Abstract. We develop an algebraic representation for ð1; 1Þ-knots using the mapping classgroup of the twice punctured torus MCG2ðTÞ. We prove that every ð1; 1Þ-knot in a lens spaceLðp; qÞ can be represented by the composition of an element of a certain rank two free sub-group of MCG2ðTÞ with a standard element only depending on the ambient space. As notableexamples, we obtain a representation of this type for all torus knots and for all two-bridgeknots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclicbranched coverings of torus knots of type ðk; ck þ 2Þ.

Key words. ð1; 1Þ-knots, Heegaard splittings, mapping class groups, two-bridge knots, torusknots.

2000 Mathematics Subject Classification. Primary 57M05, 20F38; Secondary 57M12, 57M25

1 Introduction and preliminaries

The topological properties of ð1; 1Þ-knots, also called genus one 1-bridge knots, haverecently been investigated in several papers (see [1], [5], [6], [8], [9], [10], [12], [13], [14],[15], [18], [19], [20], [21], [24], [25], [26]). These knots are very important in the lightof some results and conjectures involving Dehn surgery on knots (see in particular[9] and [25]). Moreover, the strict connections between cyclic branched coverings ofð1; 1Þ-knots and cyclic presentations of groups have been pointed out in [5], [12] and[21].

Roughly speaking, a ð1; 1Þ-knot is a knot which can be obtained by gluing alongthe boundary two solid tori with a trivial arc properly embedded. A more formaldefinition follows. A set of mutually disjoint arcs ft1; . . . ; tbg properly embedded in ahandlebody H is trivial if there exist b mutually disjoint discs D1; . . . ;Db HH suchthat ti VDi ¼ ti V qDi ¼ ti, ti VDj ¼ q and qDi � ti H qH for all i; j ¼ 1; . . . ; b andi0 j. Let M ¼ H Uj H

0 be a genus g Heegaard splitting of a closed orientable 3-manifold M and let F ¼ qH ¼ qH 0; a link LHM is said to be in b-bridge position

with respect to F if: (i) L intersects F transversally and (ii) LVH and LVH 0 are both

the union of b mutually disjoint properly embedded trivial arcs. The splitting is calleda ðg; bÞ-decomposition of L. A link L is called a ðg; bÞ-link if it admits a ðg; bÞ-decomposition. Note that a ð0; bÞ-link is a link in S3 which admits a b-bridge pre-sentation in the usual sense. So the notion of ðg; bÞ-decomposition of links in 3-manifolds generalizes the classical bridge (or plat) decomposition of links in S3 (see[7]). Obviously, a ðg; 1Þ-link is a knot, for every gd 0.

Therefore, a ð1; 1Þ-knot K is a knot in a lens space Lðp; qÞ (possibly in S3) whichadmits a ð1; 1Þ-decomposition

ðLðp; qÞ;KÞ ¼ ðH;AÞUj ðH 0;A 0Þ;

where j : ðqH 0; qA 0Þ ! ðqH; qAÞ is an (attaching) homeomorphism which reversesthe standard orientation on the tori (see Figure 1). It is well known that the familyof ð1; 1Þ-knots contains all torus knots (trivially) and all two-bridge knots (see [16])in S3.

In this paper we develop an algebraic representation of ð1; 1Þ-knots through ele-ments of MCG2ðTÞ, the mapping class group of the twice punctured torus. In Sec-tion 2 we establish the connection between the two objects. In Section 3 we provethat every ð1; 1Þ-knot in a lens space Lðp; qÞ can be represented by an element ofMCG2ðTÞ which is the composition of an element of a certain rank two free sub-group and of a standard element only depending on the ambient space Lðp; qÞ. Thisrepresentation will be called ‘‘standard’’. As a notable application, in Sections 4 and5 we obtain standard representations for the two most important classes of ð1; 1Þ-knots in S3: the torus knots and the two-bridge knots. Moreover, applying certainresults obtained in [5], we give explicit cyclic presentations for the fundamental groupsof all cyclic branched coverings of torus knots of type ðk; ck þ 2Þ, with c; k > 0 and k

odd.

ϕ

A

A'

H

H'

Figure 1. A ð1; 1Þ-decomposition.

Alessia Cattabriga and Michele Mulazzani264

In what follows, the symbol Lðp; qÞ will denote any lens space, including S3 ¼Lð1; 0Þ and S1 � S2 ¼ Lð0; 1Þ. Moreover, homotopy and homology classes will bedenoted with the same symbol of the representing loops.

2 (1, 1)-knots and MCG2(T )

Let Fg be a closed orientable surface of genus g and let P ¼ fP1; . . . ;Png be a finiteset of distinguished points of Fg, called punctures. We denote by HðFg;PÞ the groupof orientation-preserving homeomorphisms h : Fg ! Fg such that hðPÞ ¼ P. Thepunctured mapping class group of Fg relative to P is the group of the isotopy classesof elements of HðFg;PÞ. Up to isomorphism, the punctured mapping class groupof a fixed surface Fg relative to P only depends on the cardinality n of P. Therefore,we can simply speak of the n-punctured mapping class group of Fg, denoting it byMCGnðFgÞ. Moreover, for isotopy classes we will use the same symbol of the repre-senting homeomorphisms.

The n-punctured pure mapping class group of Fg is the subgroup PMCGnðFgÞ ofMCGnðFgÞ consisting of the elements pointwise fixing the punctures. There is astandard exact sequence

1 ! PMCGnðFgÞ ! MCGnðFgÞ ! Sn ! 1;

where Sn is the symmetric group on n elements. A presentation of all puncturedmapping class groups can be found in [11] and in [17].

In this paper we are interested in the two-punctured mapping class group of thetorus MCG2ðTÞ. According to previously cited papers, a set of generators forMCG2ðTÞ is given by a rotation r of p radians which exchanges the punctures andthe right-handed Dehn twists ta; tb; tg around the curves a; b; g respectively, as de-picted in Figure 2. Since r commutes with the other generators, we have

MCG2ðTÞGPMCG2ðTÞlZ2:

T

P1 P2

α

g

b

Figure 2. Generators of MCG2ðTÞ.

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 265

The following presentation for PMCG2ðTÞ has been obtained in [22]:

hta; tb; tg j tatbta ¼ tbtatb; tatgta ¼ tgtatg; tbtg ¼ tgtb; ðtatbtgÞ4 ¼ 1i: ð1Þ

The group PMCG2ðTÞ (as well as MCG2ðTÞ) naturally maps by an epimorphismto the mapping class group of the torus MCGðTÞG SLð2;ZÞ, which is generated byta and tb ¼ tg. So we have an epimorphism

W : PMCG2ðTÞ ! SLð2;ZÞ

defined by WðtaÞ ¼1 0

1 1

� �and WðtbÞ ¼ WðtgÞ ¼

1 �1

0 1

� �.

The group kerW will play a fundamental role in our discussion. In order to inves-tigate its structure, let us consider the two elements tm ¼ tbt

�1g and tl ¼ tht

�1a , where

th is the right-handed Dehn twist around the curve h depicted in Figure 3. The e¤ectof tm and tl is to slide one puncture (say P2) respectively along a meridian and alonga longitude of the torus, as shown in Figure 3. Observe that, since h ¼ t�1

m ðaÞ, wehave th ¼ t�1

m tatm.The following result can be obtained from [3, Theorem 1] and [2, Theorem 5] by

classical techniques.

Proposition 1. The group kerW is freely generated by tm ¼ tbt�1g and tl ¼ tht

�1a , where

th ¼ t�1m tatm.

Now, let KHLðp; qÞ be a ð1; 1Þ-knot with ð1; 1Þ-decomposition ðLðp; qÞ;KÞ ¼ðH;AÞUj ðH 0;A 0Þ and let m : ðH;AÞ ! ðH 0;A 0Þ be a fixed orientation-reversinghomeomorphism, then c ¼ jmjqH is an orientation-preserving homeomorphism ofðqH; qAÞ ¼ ðT ; fP1;P2gÞ. Moreover, since two isotopic attaching homeomorphisms

m

l

T T

T T

g b

α

h

Figure 3. Action of tm and tl .

Alessia Cattabriga and Michele Mulazzani266

produce equivalent ð1; 1Þ-knots, we have a natural surjective map from the twicepunctured mapping class group of the torus MCG2ðTÞ to the class K1;1 of all ð1; 1Þ-knots

Y : c A MCG2ðTÞ 7! Kc A K1;1:

If WðcÞ ¼ q s

p r

� �, then Kc is a ð1; 1Þ-knot in the lens space Lðjpj; jqjÞ [4, p. 186],

and therefore it is a knot in S3 if and only if p ¼G1.As will be proved in Section 3, we have the following ‘‘trivial’’ examples:

i) if either c ¼ 1 or c ¼ tb or c ¼ tg, then Kc is the trivial knot in S1 � S2;

ii) if c ¼ ta, then Kc is the trivial knot in S3.

Moreover, it is possible to prove that if c ¼ tatbtatatgta, then Kc is the knot

S1 � fPgHS1 � S2, where P is any point of S2. So, in this case, Kc is a standardgenerator for the first homology group of S1 � S2.

Every element c of MCG2ðTÞ can be written as c ¼ c 0rk, k A f0; 1g, wherec 0 A PMCG2ðTÞ. Since r can be extended to a homeomorphism of the pair ðH;AÞ,the ð1; 1Þ-knots Kc and Kc 0 are equivalent. So, for our discussion it is enough toconsider the restriction

Y 0 ¼ YjPMCG2ðTÞ : c A PMCG2ðTÞ 7! Kc A K1;1:

3 Standard decomposition

In this section we show that every ð1; 1Þ-knot KHLðp; qÞ admits a representationby the composition of an element in kerW and an element which only depends onLðp; qÞ. A representation of this type will be called ‘‘standard’’. Note that a similarresult, using a rank three free subgroup of MCG2ðTÞ, has been obtained in [6, The-orem 3].

First of all, we deal with trivial knots in lens spaces. Let T be the subgroup ofPMCG2ðTÞ generated by ta and tb. There exists a disk DHH, with AVD ¼AV qD ¼ A and qD� AHT , such that DV a ¼ DV b ¼ q. So any element of Tproduces a trivial knot in a certain lens space. On the other hand, any trivial knot ina lens space admits a representation through an element of T, as will be proved inProposition 3.

We need a preparatory result.

Lemma 2. Let K be a ð1; 1Þ-knot in Lðp; qÞ. Then, for each r; s A Z such that

qr� ps ¼ 1 there exists c A PMCG2ðTÞ, with WðcÞ ¼ q s

p r

� �, such that K ¼ Kc.

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 267

Proof. Let K ¼ Kc

, with WðcÞ ¼ q s

p r

� �. Since qr� ps ¼ 1, there exist c A Z such

that r ¼ rþ cp and s ¼ sþ cq. If c ¼ ct�cb , we have Kc ¼ K

c, since t�c

b can be ex-tended to a homeomorphism of the pair ðH;AÞ. Moreover WðcÞ ¼ WðcÞWðt�c

b Þ ¼q s

p r

� �1 c

0 1

� �¼ q sþ cq

p rþ cp

� �. r

For integers p; q such that 0 < q < p and gcdðp; qÞ ¼ 1 consider the sequence ofequations of the Euclidean algorithm (with r0 ¼ p, r1 ¼ q):

r0 ¼ a1r1 þ r2

r1 ¼ a2r2 þ r3

..

.

rm�2 ¼ am�1rm�1 þ rm

rm�1 ¼ amrm;

with r1 > r2 > � � � > rm�1 > rm ¼ 1.The ai’s are the coe‰cients of the continued fraction

p

q¼ a1 þ

1

a2 þ 1a3þ���þ 1

am

:

In the following we will use the notation p=q ¼ ½a1; a2; . . . ; am�.

Proposition 3. The trivial knot in S3 ¼ Lð1; 0Þ is represented by c1;0 ¼ tbtatb.The trivial knot in S1 � S2 ¼ Lð0; 1Þ is represented by c0;1 ¼ 1.Let p; q be integers such that 0 < q < p and gcdðp; qÞ ¼ 1. If p=q ¼ ½a1; a2; . . . ; am�,

then the trivial knot in the lens space Lðp; qÞ is represented by

cp;q ¼ta1a t�a2

b . . . tama if m is odd;

ta1a t�a2

b . . . t�amb tbtatb if m is even:

(

Proof. Since all the involved homeomorphisms belong to T, all the knots are trivial.It is easy to check (see also [4, p. 186]) that, for suitable r; s A Z, we have:

q s

p r

� �¼

1 0

a1 1

� �1 a2

0 1

� �. . .

1 0

am 1

� �if m is odd;

1 0

a1 1

� �1 a2

0 1

� �. . .

1 am

0 1

� �0 �1

1 0

� �if m is even.

8>>><>>>:

Alessia Cattabriga and Michele Mulazzani268

Since Wðtaia Þ ¼1 0

ai 1

� �, Wðtaib Þ ¼

1 �ai

0 1

� �, and WðtbtatbÞ ¼

0 �1

1 0

� �, the state-

ment is obtained. r

Now we can prove the result announced at the beginning of this section.

Theorem 4. Let K be a ð1; 1Þ-knot in Lðp; qÞ. Then there exist c 0;c 00 A kerW such that

K ¼ Kc, with c ¼ c 0cp;q ¼ cp;qc00.

Proof. By Lemma 2, there exists c, with WðcÞ ¼ Wðcp;qÞ, such that K ¼ Kc. It suf-fices to define c 0 ¼ cc�1

p;q and c 00 ¼ c�1p;qc. r

A representation c A PMCG2ðTÞ of a ð1; 1Þ-knot will be called standard if c is ofthe type described in the previous theorem.

We point out that ð1; 1Þ-knots admit di¤erent (usually infinitely many) standardrepresentations. For example, tcm represents the trivial knot in S1 � S2, for allc A Z.

4 Representation of torus knots

In this section we give a standard representation for all torus knots in S3. LetK ¼ tðk; hÞ be a torus knot of type ðk; hÞ. Then gcdðk; hÞ ¼ 1, and we can assumethat K lies on the boundary T ¼ qH of a genus one handlebody H canonically em-bedded in S3. The homology class of K is hl þ km, where l and m respectively denotea longitude and a meridian of T . By slightly pushing (the interior of ) an arc A 0 HK

outside H and K � A 0 inside H, we obtain an obvious ð1; 1Þ-decomposition of K .Observe that 0 < jkj < h can be assumed without loss of generality (see [4, p. 45]).

In the next statement bxc denotes the integral part of x.

Theorem 5. The torus knot tðk; hÞHS3 is the ð1; 1Þ-knot Kc with:

c ¼Yhi¼1

ðtbði�1Þk=hc�bik=hcm t�1

l Þtbtatb;

where tm ¼ tbt�1g and tl ¼ t�1

m tatmt�1a .

Proof. Up to isotopy, we can suppose that the arc A ¼ Kc � intðA 0Þ lies on qH, asin Figure 4. The arc A can be transformed into an arc ~AA in such a way that ~AAUA 0 isa trivial knot in S3, represented by the standard homeomorphism c1;0 ¼ tbtatb, via asuitable sequence of homeomorphisms tl and tm, according to the following algo-rithm. Consider the sequence of equations:

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 269

k ¼ q1hþ r1;

2k ¼ q2hþ r2;

..

.

hk ¼ qhhþ rh;

where 0c ri < h, for i ¼ 1; . . . ; h. Moreover, define q0 ¼ 0. So qi ¼ bik=hc, for i ¼ 0;1; . . . ; h. Now define the homeomorphisms ci ¼ tlt

qi�qi�1m , for i ¼ 1; . . . ; h. Figure 5

depicts the e¤ect of tl and tltm on A. As a consequence, the homeomorphism f ¼chch�1 . . .c1 transforms the arc A into the arc ~AA (Figure 6 shows the case tð5; 7Þ),and therefore we have c1;0 ¼ fc. So f�1c1;0 represents the torus knot tðk; hÞ. r

For example, tð5; 7Þ ¼ Kc, with c ¼ t�1l ðt�1

m t�1l Þ2t�1

l ðt�1m t�1

l Þ3tbtatb (see Figure 6).

As a consequence, we obtain a cyclic presentation for the fundamental group forall cyclic branched coverings of a particular class of torus knots.

H

A

A'

Figure 4.

l

l m

Figure 5.

Alessia Cattabriga and Michele Mulazzani270

Proposition 6. The fundamental group of the n-fold cyclic branched covering of the

torus knot tðk; ck þ 2Þ, with k > 1 odd and c > 0, admits the cyclic presentation GnðwÞ,where w is equal to

Yðk�3Þ=2

i¼0

� Ycðk�1Þ=2

j¼0

x1�iðckþ2Þþjk

Ycðkþ1Þ=2

l¼0

x�1ckðk�1Þ=2�iðckþ2Þ�lk

� Ycðk�1Þ=2

m¼0

x1�ðk�1Þðckþ2Þ=2þmk

(subscripts are taken modulo n).

l

l m

l m

l

l m

l m

l m

Figure 6. Trivialization of tð5; 7Þ.

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 271

Proof. Let r ¼ ðk � 1Þ=2. From Theorem 5 we have tðk; ck þ 2Þ ¼ Kc withc ¼ ðt�c

l t�1m Þrt�1

l ðt�cl t�1

m Þ rt�cl t�1

m t�1l tbtatb. Applying [5, Proposition 1], we obtain

p1ðS3 � tðk; ck þ 2ÞÞ ¼ ha; g j rða; gÞi, with rða; gÞ ¼ ðg�1acrþ1g�1a�cðrþ1Þ�1Þrg�1acrþ1.Then H1ðS3 � tðk; ck þ 2ÞÞ ¼ ha; g j a� kgi. Since, up to equivalence, of ðgÞ ¼ 1, wehave of ðaÞ ¼ k. We set a ¼ xgk, therefore p1ðS3 � tðk; ck þ 2ÞÞ ¼ hx; g j rðx; gÞi,

with rðx; gÞ ¼ ðg�1ðxgkÞ1þcðk�1Þ=2g�1ðg�kx�1Þ1þcðkþ1Þ=2Þðk�1Þ=2g�1ðxgkÞ1þcðk�1Þ=2. Thestatement derives from a straightforward application of [5, Theorem 7]. r

For example, the fundamental group of the n-fold cyclic branched covering oftð5; 7Þ admits the cyclic presentation GnðwÞ, where

w ¼ x15x20x25x�124 x

�119 x

�114 x

�19 x8x13x18x

�117 x

�112 x

�17 x�1

2 x1x6x11:

5 Representation of two-bridge knots

In this section we give a standard representation for all two-bridge knots in S3. Letbða=bÞ be a non-trivial two-bridge knot in S3 of type ða; bÞ. Then we can assumegcdða; bÞ ¼ 1, a odd, b even and 0 < jbj < a, without loss of generality (see [4, Ch.12B]). It is known that bða=bÞ admits a Conway presentation with an even numberof even parameters ½2a1; 2b1; . . . ; 2an; 2bn� (see Figure 7), satisfying the following re-lation:

a

b¼ 2a1 þ

1

2b1 þ 12a2þ���þ 1

2bn

:

Theorem 7. The two-bridge knot bða=bÞHS3 having Conway parameters ½2a1; 2b1; . . . ;2an; 2bn� is the ð1; 1Þ-knot Kc with:

c ¼ tbtatbt�bnm tane . . . t�b1

m ta1e ;

where te ¼ t�1l tmtlt

�1m is the right-handed Dehn twist around the curve e depicted in

Figure 8.

Proof. Figure 8 shows the result of the application of t�bnm tane . . . t�b1

m ta1e . By applying

c1;0 ¼ tbtatb we obtain the two-bridge knot with Conway parameters ½2a1; 2b1; . . . ;2an; 2bn�.

2a1

2b12an

n2b

Figure 7. Conway presentation for two-bridge knots.

Alessia Cattabriga and Michele Mulazzani272

εe-bnm

an -bm

2 a2t

2a1

2b1

2bn

2an

2b1

2a1

A

e

12a

b

g

eta1

m1-b

t

Figure 8. Standard representation of two-bridge knots.

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 273

Now we show that te ¼ t�1l tmtlt

�1m (note that no disk bounded by e and properly

embedded in H is disjoint from A). Referring to Figure 9, the following ‘‘lantern’’relation t2

g td1td2

¼ tetbtz holds (see [23]). So we obtain z ¼ tatgt�1b t�1

a ðgÞ and therefore

tz ¼ tatgt�1b t�1

a tgtatbt�1g t�1

a . Since td1¼ td2

¼ 1 we have te ¼ t2g t

�1z t�1

b ¼ t2g tatgt

�1b t�1

a t�1g �

tatbt�1g t�1

a t�1b . Now, using the relations of (1) we get

te ¼ t2g tatgt

�1b t�1

a t�1g tatbt

�1g t�1

a t�1b ¼ tgtatgtat

�1b t�1

a t�1g tatbt

�1g t�1

a t�1b

¼ tgtatgt�1b t�1

a tbt�1g tatbt

�1g t�1

a t�1b ¼ tgtat

�1b tgt

�1a t�1

g tbtatbt�1g t�1

a t�1b

¼ tgtat�1b t�1

a t�1g tatbtatbt

�1g t�1

a t�1b ¼ tgt

�1b t�1

a tbt�1g tatbtatbt

�1g t�1

a t�1b

¼ tgt�1b t�1

a tbt�1g tatatbtat

�1g t�1

a t�1b ¼ tgt

�1b t�1

a tbt�1g tatatbt

�1g t�1

a tgt�1b

¼ t�1m t�1

a tmtatatmt�1a t�1

m ¼ t�1m t�1

a tmtatmt�1m tatmt

�1a t�1

m

¼ t�1h tatmtht

�1a t�1

m ¼ t�1l tmtlt

�1m : r

For example, the figure-eight knot bð5=2Þ, which has Conway parameters ½2; 2�, is theknot Kc with c ¼ tbtatbt

�1m te (see Figure 10).

Acknowledgements. The authors would like to thank Sylvain Gervais and AndreiVesnin for their helpful suggestions. We also would like to thank the Referee forhis valuable comments and remarks. Work performed under the auspices of theG.N.S.A.G.A. of I.N.d.A.M. (Italy) and the University of Bologna, funds for se-lected research topics.

T

g g

d1e zd2b

A

Figure 9.

Alessia Cattabriga and Michele Mulazzani274

References

[1] J. Berge, The knots in D2 � S1 which have nontrivial Dehn surgeries that yield D2 � S1.Topology Appl. 38 (1991), 1–19. MR 92d:57005 Zbl 0725.57001

[2] J. S. Birman, On braid groups. Comm. Pure Appl. Math. 21 (1968), 41–72. MR 38 #2764Zbl 0157.30904

et

m-1

Figure 10. Standard representation of the figure-eight knot.

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 275

[3] J. S. Birman, Mapping class groups and their relationship to braid groups. Comm. Pure

Appl. Math. 22 (1969), 213–238. MR 39 #4840 Zbl 0167.21503[4] G. Burde, H. Zieschang, Knots, volume 5 of de Gruyter Studies in Mathematics. de

Gruyter 1985. MR 87b:57004 Zbl 0568.57001[5] A. Cattabriga, M. Mulazzani, Strongly-cyclic branched coverings of ð1; 1Þ-knots and

cyclic presentations of groups. Math. Proc. Cambridge Philos. Soc. 135 (2003), 137–146.MR 1 990 837

[6] D. H. Choi, K. H. Ko, Parametrizations of 1-bridge torus knots. J. Knot Theory Ram-

ifications 12 (2003), 463–491. MR 1 985 906[7] H. Doll, A generalized bridge number for links in 3-manifolds. Math. Ann. 294 (1992),

701–717. MR 93i:57023 Zbl 0757.57012[8] H. Fujii, Geometric indices and the Alexander polynomial of a knot. Proc. Amer. Math.

Soc. 124 (1996), 2923–2933. MR 96m:57014 Zbl 0861.57012[9] D. Gabai, Surgery on knots in solid tori. Topology 28 (1989), 1–6. MR 90h:57005

Zbl 0678.57004[10] D. Gabai, 1-bridge braids in solid tori. Topology Appl. 37 (1990), 221–235.

MR 92b:57011 Zbl 0817.57006[11] S. Gervais, A finite presentation of the mapping class group of a punctured surface. To-

pology 40 (2001), 703–725. MR 2002m:57025 Zbl 0992.57013[12] L. Grasselli, M. Mulazzani, Genus one 1-bridge knots and Dunwoody manifolds. Forum

Math. 13 (2001), 379–397. MR 2002g:57003 Zbl 0963.57002[13] C. Hayashi, Genus one 1-bridge positions for the trivial knot and cabled knots. Math.

Proc. Cambridge Philos. Soc. 125 (1999), 53–65. MR 99j:57005 Zbl 0961.57004[14] C. Hayashi, Satellite knots in 1-genus 1-bridge positions. Osaka J. Math. 36 (1999),

711–729. MR 2001a:57009 Zbl 0953.57003[15] C. Hayashi, 1-genus 1-bridge splittings for knots in the 3-sphere and lens spaces. Preprint.[16] T. Kobayashi, O. Saeki, The Rubinstein-Scharlemann graphic of a 3-manifold as the dis-

criminant set of a stable map. Pacific J. Math. 195 (2000), 101–156. MR 2001i:57026Zbl 01537853

[17] C. Labruere, L. Paris, Presentations for the punctured mapping class groups in terms ofArtin groups. Algebr. Geom. Topol. 1 (2001), 73–114. MR 2002a:57003 Zbl 0962.57008

[18] K. Morimoto, M. Sakuma, On unknotting tunnels for knots. Math. Ann. 289 (1991),143–167. MR 92e:57015 Zbl 0697.57002

[19] K. Morimoto, M. Sakuma, Y. Yokota, Examples of tunnel number one knots which havethe property ‘‘1 þ 1 ¼ 3’’. Math. Proc. Cambridge Philos. Soc. 119 (1996), 113–118.MR 96i:57007 Zbl 0866.57004

[20] K. Morimoto, M. Sakuma, Y. Yokota, Identifying tunnel number one knots. J. Math.

Soc. Japan 48 (1996), 667–688. MR 97g:57010 Zbl 0869.57008[21] M. Mulazzani, Cyclic presentations of groups and cyclic branched coverings of ð1; 1Þ-

knots. Bull. Korean Math. Soc. 40 (2003), 101–108. MR 1 958 228[22] J. R. Parker, C. Series, The mapping class group of the twice punctured torus. To appear

in Proceedings of the conference ‘‘Groups: Combinatorial and Geometric Aspects’’ (Biele-feld, 15–23 August 1999), London Mathematical Society Lecture Note Series.

[23] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface.Israel J. Math. 45 (1983), 157–174. MR 85g:57007 Zbl 0533.57002

[24] Y. Q. Wu, Incompressibility of surfaces in surgered 3-manifolds. Topology 31 (1992),271–279. MR 94e:57027 Zbl 0872.57022

[25] Y. Q. Wu, q-reducing Dehn surgeries and 1-bridge knots. Math. Ann. 295 (1993), 319–331.MR 94a:57036 Zbl 0788.57005

Alessia Cattabriga and Michele Mulazzani276

[26] Y.-Q. Wu, Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebodies.Math. Proc. Cambridge Philos. Soc. 120 (1996), 687–696. MR 97i:57021 Zbl 0888.57015

Received 9 September, 2002; revised 12 December, 2002

A. Cattabriga, Department of Mathematics, University of Bologna, ItalyEmail: cattabri@dm.unibo.it

M. Mulazzani, Department of Mathematics and C.I.R.A.M., University of Bologna, ItalyEmail: mulazza@dm.unibo.it

ð1; 1Þ-knots via the mapping class group of the twice punctured torus 277