(de Gruyter 2004
(1, 1)-knots via the mapping class group of the 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 class group of the twice punctured torus MCG2ðTÞ. We prove that everyð1;1Þ-knot in a lens space Lðp;qÞcan be represented by the composition of an element of a certain rank two free sub- group of MCG2ðTÞwith a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of typeðk;ckþ2Þ.
Key words. ð1;1Þ-knots, Heegaard splittings, mapping class groups, two-bridge knots, torus knots.
2000 Mathematics Subject Classification. Primary 57M05, 20F38; Secondary 57M12, 57M25
1 Introduction and preliminaries
The topological properties ofð1;1Þ-knots, also called genus one 1-bridge knots, have recently 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 light of some results and conjectures involving Dehn surgery on knots (see in particular [9] and [25]). Moreover, the strict connections between cyclic branched coverings of ð1;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 along the boundary two solid tori with a trivial arc properly embedded. A more formal definition follows. A set of mutually disjoint arcsft1;. . .;tbgproperly embedded in a handlebody H is trivialif there existb mutually disjoint discsD1;. . .;DbHH such that tiVDi¼tiVqDi¼ti, tiVDj¼qand qDitiHqH for alli;j¼1;. . .;b and i0j. Let M¼HUjH0 be a genus g Heegaard splitting of a closed orientable 3- manifoldM and letF ¼qH ¼qH0; a linkLHM is said to be inb-bridge position with respect toF if: (i)LintersectsF transversally and (ii)LVHandLVH0are both
the union ofbmutually disjoint properly embedded trivial arcs. The splitting is called a 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 inS3 (see [7]). Obviously, aðg;1Þ-link is a knot, for everygd0.
Therefore, að1;1Þ-knot K is a knot in a lens spaceLðp;qÞ(possibly inS3) which admits að1;1Þ-decomposition
ðLðp;qÞ;KÞ ¼ ðH;AÞUjðH0;A0Þ;
where j:ðqH0;qA0Þ ! ðqH;qAÞ is an (attaching) homeomorphism which reverses the standard orientation on the tori (see Figure 1). It is well known that the family ofð1;1Þ-knots contains all torus knots (trivially) and all two-bridge knots (see [16]) inS3.
In this paper we develop an algebraic representation ofð1;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 prove that every ð1;1Þ-knot in a lens space Lðp;qÞ can be represented by an element of MCG2ðTÞ 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 spaceLðp;qÞ. This representation will be called ‘‘standard’’. As a notable application, in Sections 4 and 5 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 certain results obtained in [5], we give explicit cyclic presentations for the fundamental groups of all cyclic branched coverings of torus knots of typeðk;ckþ2Þ, withc;k>0 andk odd.
ϕ
A
A'
H
H'
Figure 1. Að1;1Þ-decomposition.
In what follows, the symbol Lðp;qÞ will denote any lens space, including S3 ¼ Lð1;0Þ and S1S2¼Lð0;1Þ. Moreover, homotopy and homology classes will be denoted with the same symbol of the representing loops.
2 (1, 1)-knots and MCG2(T)
LetFg be a closed orientable surface of genusg and letP¼ fP1;. . .;Pngbe a finite set of distinguished points ofFg, calledpunctures. We denote byHðFg;PÞthe group of orientation-preserving homeomorphisms h:Fg!Fg such that hðPÞ ¼P. The punctured mapping class groupofFg relative toPis the group of the isotopy classes of elements of HðFg;PÞ. Up to isomorphism, the punctured mapping class group of a fixed surfaceFg relative toPonly depends on the cardinalitynofP. Therefore, we can simply speak of the n-punctured mapping class group of Fg, denoting it by MCGnðFgÞ. 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 of MCGnðFgÞ consisting of the elements pointwise fixing the punctures. There is a standard exact sequence
1!PMCGnðFgÞ !MCGnðFgÞ !Sn!1;
where Sn is the symmetric group on n elements. A presentation of all punctured mapping class groups can be found in [11] and in [17].
In this paper we are interested in the two-punctured mapping class group of the torus MCG2ðTÞ. According to previously cited papers, a set of generators for MCG2ðTÞis given by a rotation rofpradians which exchanges the punctures and the right-handed Dehn twists ta;tb;tg around the curves a;b;g respectively, as de- picted in Figure 2. Sincercommutes with the other generators, we have
MCG2ðTÞGPMCG2ðTÞlZ2: T
P1 P2
α g b
Figure 2. Generators of MCG2ðTÞ.
The following presentation for PMCG2ðTÞhas been obtained in [22]:
hta;tb;tgjtatbta¼tbtatb;tatgta¼tgtatg;tbtg¼tgtb;ðtatbtgÞ4¼1i: ð1Þ The group PMCG2ðTÞ(as well as MCG2ðTÞ) naturally maps by an epimorphism to the mapping class group of the torus MCGðTÞGSLð2;ZÞ, which is generated by taandtb ¼tg. So we have an epimorphism
W:PMCG2ðTÞ !SLð2;ZÞ defined byWðtaÞ ¼ 1 0
1 1
andWðtbÞ ¼WðtgÞ ¼ 1 1
0 1
.
The group kerWwill play a fundamental role in our discussion. In order to inves- tigate its structure, let us consider the two elementstm¼tbtg1 andtl¼tht1a , where this the right-handed Dehn twist around the curvehdepicted in Figure 3. The e¤ect oftmandtl is to slide one puncture (sayP2) respectively along a meridian and along a longitude of the torus, as shown in Figure 3. Observe that, since h¼t1m ðaÞ, we haveth¼t1m tatm.
The following result can be obtained from [3, Theorem 1] and [2, Theorem 5] by classical techniques.
Proposition 1.The groupkerWis freely generated bytm¼tbtg1 andtl¼tht1a ,where th¼t1m tatm.
Now, let KHLðp;qÞ be a ð1;1Þ-knot with ð1;1Þ-decomposition ðLðp;qÞ;KÞ ¼ ðH;AÞUjðH0;A0Þ and let m:ðH;AÞ ! ðH0;A0Þ be a fixed orientation-reversing homeomorphism, then c¼jmjqH is an orientation-preserving homeomorphism of ðqH;qAÞ ¼ ðT;fP1;P2gÞ. Moreover, since two isotopic attaching homeomorphisms
m
l
T T
T T
g b
α h
Figure 3. Action oftmandtl.
produce equivalent ð1;1Þ-knots, we have a natural surjective map from the twice punctured mapping class group of the torus MCG2ðTÞto the classK1;1of allð1;1Þ- knots
Y:cAMCG2ðTÞ 7!KcAK1;1:
IfWðcÞ ¼ q s p r
, thenKcis að1;1Þ-knot in the lens spaceLðjpj;jqjÞ[4, p. 186], and therefore it is a knot inS3if and only if p¼G1.
As will be proved in Section 3, we have the following ‘‘trivial’’ examples:
i) if eitherc¼1 orc¼tb orc¼tg, thenKcis the trivial knot inS1S2; ii) ifc¼ta, thenKcis the trivial knot inS3.
Moreover, it is possible to prove that if c¼tatbtatatgta, then Kc is the knot S1 fPgHS1S2, where P is any point ofS2. So, in this case, Kc is a standard generator for the first homology group ofS1S2.
Every element c of MCG2ðTÞ can be written as c¼c0rk, kAf0;1g, where c0APMCG2ðTÞ. Since r can be extended to a homeomorphism of the pairðH;AÞ, the ð1;1Þ-knots Kc and Kc0 are equivalent. So, for our discussion it is enough to consider the restriction
Y0¼YjPMCG2ðTÞ:cAPMCG2ðTÞ 7!KcAK1;1:
3 Standard decomposition
In this section we show that every ð1;1Þ-knot KHLðp;qÞ admits a representation by the composition of an element in kerWand an element which only depends on Lðp;qÞ. A representation of this type will be called ‘‘standard’’. Note that a similar result, 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 of PMCG2ðTÞ generated by ta and tb. There exists a disk DHH, with AVD¼ AVqD¼Aand qDAHT, such that DVa¼DVb¼q. So any element ofT produces a trivial knot in a certain lens space. On the other hand, any trivial knot in a lens space admits a representation through an element ofT, as will be proved in Proposition 3.
We need a preparatory result.
Lemma 2. Let K be a ð1;1Þ-knot in Lðp;qÞ. Then, for each r;sAZ such that qrps¼1there existscAPMCG2ðTÞ,withWðcÞ ¼ q s
p r
,such that K ¼Kc.
Proof.Let K¼Kc, withWðcÞ ¼ q s p r
. Sinceqrps¼1, there existcAZsuch that r¼rþcpands¼sþcq. Ifc¼ctbc, we haveKc¼Kc, since tbc can be ex- tended to a homeomorphism of the pair ðH;AÞ. Moreover WðcÞ ¼WðcÞWðtbcÞ ¼
q s p r
1 c 0 1
¼ q sþcq p rþcp
. r
For integers p;q such that 0<q<p and gcdðp;qÞ ¼1 consider the sequence of equations of the Euclidean algorithm (withr0¼p,r1¼q):
r0¼a1r1þr2
r1¼a2r2þr3
...
rm2¼am1rm1þrm rm1¼amrm;
withr1>r2 > >rm1>rm¼1.
Theai’s are the coe‰cients of the continued fraction p
q ¼a1þ 1 a2þ 1
a3þþam1
:
In the following we will use the notation p=q¼ ½a1;a2;. . .;am.
Proposition 3.The trivial knot inS3¼Lð1;0Þis represented byc1;0¼tbtatb. The trivial knot inS1S2¼Lð0;1Þis represented byc0;1¼1.
Let p;q be integers such that0<q< p andgcdðp;qÞ ¼1.If p=q¼ ½a1;a2;. . .;am, then the trivial knot in the lens space Lðp;qÞis represented by
cp;q ¼ taa1tab 2. . .taam if m is odd;
taa1tab 2. . .tab mtbtatb if m is even:
(
Proof.Since all the involved homeomorphisms belong toT, all the knots are trivial.
It is easy to check (see also [4, p. 186]) that, for suitabler;sAZ, we have:
q s p r
¼
1 0
a1 1
1 a2
0 1
. . . 1 0
am 1
if mis odd;
1 0
a1 1
1 a2
0 1
. . . 1 am
0 1
0 1
1 0
if mis even.
8>
>>
<
>>
>:
SinceWðtaaiÞ ¼ 1 0 ai 1
,WðtbaiÞ ¼ 1 ai
0 1
, andWðtbtatbÞ 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 að1;1Þ-knot in Lðp;qÞ.Then there existc0;c00AkerWsuch that K¼Kc,withc¼c0cp;q¼cp;qc00.
Proof. By Lemma 2, there existsc, withWðcÞ ¼Wðcp;qÞ, such thatK¼Kc. It suf-
fices to definec0¼cc1p;qandc00¼c1p;qc. r
A representationcAPMCG2ðTÞof að1;1Þ-knot will be calledstandardifcis of the type described in the previous theorem.
We point out that ð1;1Þ-knots admit di¤erent (usually infinitely many) standard representations. For example, tmc represents the trivial knot in S1S2, for all cAZ.
4 Representation of torus knots
In this section we give a standard representation for all torus knots in S3. Let K¼tðk;hÞ be a torus knot of type ðk;hÞ. Then gcdðk;hÞ ¼1, and we can assume thatK lies on the boundaryT ¼qH of a genus one handlebodyH canonically em- bedded inS3. The homology class ofKishlþkm, wherelandmrespectively denote a longitude and a meridian ofT. By slightly pushing (the interior of ) an arcA0HK outside H and KA0 inside H, we obtain an obviousð1;1Þ-decomposition of K.
Observe that 0<jkj<hcan be assumed without loss of generality (see [4, p. 45]).
In the next statementbxcdenotes the integral part ofx.
Theorem 5.The torus knottðk;hÞHS3is theð1;1Þ-knot Kc with:
c¼Yh
i¼1
ðtbði1Þk=hcbik=hc
m t1l Þtbtatb;
wheretm¼tbtg1 andtl¼t1m tatmta1.
Proof. Up to isotopy, we can suppose that the arcA¼KcintðA0Þ lies onqH, as in Figure 4. The arcAcan be transformed into an arcAA~in such a way thatAA~UA0is a trivial knot inS3, represented by the standard homeomorphismc1;0¼tbtatb, via a suitable sequence of homeomorphisms tl and tm, according to the following algo- rithm. Consider the sequence of equations:
k¼q1hþr1; 2k¼q2hþr2;
...
hk¼qhhþrh;
where 0cri<h, fori¼1;. . .;h. Moreover, defineq0¼0. Soqi¼ bik=hc, fori¼0;
1;. . .;h. Now define the homeomorphismsci¼tltmqiqi1, for i¼1;. . .;h. Figure 5
depicts the e¤ect of tl and tltm on A. As a consequence, the homeomorphismf¼
chch1. . .c1 transforms the arc Ainto the arcAA~(Figure 6 shows the casetð5;7Þ),
and therefore we havec1;0¼fc. Sof1c1;0represents the torus knottðk;hÞ. r For example,tð5;7Þ ¼Kc, withc¼t1l ðt1m t1l Þ2t1l ðt1m t1l Þ3tbtatb(see Figure 6).
As a consequence, we obtain a cyclic presentation for the fundamental group for all cyclic branched coverings of a particular class of torus knots.
H A
A' Figure 4.
l
l m
Figure 5.
Proposition 6. The fundamental group of the n-fold cyclic branched covering of the torus knottðk;ckþ2Þ,with k>1odd and c>0,admits the cyclic presentation GnðwÞ, where w is equal to
ðk3Þ=2Y
i¼0
Ycðk1Þ=2
j¼0
x1iðckþ2Þþjk cðkþ1Þ=2Y
l¼0
x1ckðk1Þ=2iðckþ2Þlk
Ycðk1Þ=2
m¼0
x1ðk1Þðckþ2Þ=2þmk
(subscripts are taken modulo n).
l
l m
l m
l
l m
l m
l m
Figure 6. Trivialization oftð5;7Þ.
Proof. Let r¼ ðk1Þ=2. From Theorem 5 we have tðk;ckþ2Þ ¼Kc with c¼ ðtcl t1m Þrt1l ðtcl t1m Þrtcl t1m t1l tbtatb. Applying [5, Proposition 1], we obtain p1ðS3tðk;ckþ2ÞÞ ¼ha;gjrða;gÞi, withrða;gÞ ¼ ðg1acrþ1g1acðrþ1Þ1Þrg1acrþ1. ThenH1ðS3tðk;ckþ2ÞÞ ¼ha;gjakgi. Since, up to equivalence,ofðgÞ ¼1, we have ofðaÞ ¼k. We set a¼xgk, therefore p1ðS3tðk;ckþ2ÞÞ ¼hx;gjrðx;gÞi, with rðx;gÞ ¼ ðg1ðxgkÞ1þcðk1Þ=2g1ðgkx1Þ1þcðkþ1Þ=2Þðk1Þ=2g1ðxgkÞ1þcðk1Þ=2. The statement derives from a straightforward application of [5, Theorem 7]. r For example, the fundamental group of the n-fold cyclic branched covering of tð5;7Þadmits the cyclic presentationGnðwÞ, where
w¼x15x20x25x124x119x114x19 x8x13x18x117x112x17 x12 x1x6x11:
5 Representation of two-bridge knots
In this section we give a standard representation for all two-bridge knots inS3. Let bða=bÞ be a non-trivial two-bridge knot in S3 of type ða;bÞ. Then we can assume gcdða;bÞ ¼1,a odd, b even and 0<jbj<a, without loss of generality (see [4, Ch.
12B]). It is known thatbða=bÞadmits a Conway presentation with an even number of even parameters½2a1;2b1;. . .;2an;2bn(see Figure 7), satisfying the following re- lation:
a
b¼2a1þ 1 2b1þ 1
2a2þþ1
2bn
:
Theorem 7.The two-bridge knotbða=bÞHS3having Conway parameters½2a1;2b1;. . .; 2an;2bnis theð1;1Þ-knot Kc with:
c¼tbtatbtbm ntean. . .tbm1tea1;
where te¼t1l tmtlt1m is the right-handed Dehn twist around the curve edepicted in Figure8.
Proof.Figure 8 shows the result of the application of tbm ntean. . .tbm1tea1. By applying c1;0¼tbtatb we obtain the two-bridge knot with Conway parameters ½2a1;2b1;. . .; 2an;2bn.
2a1
2b1
2an n
2b
Figure 7. Conway presentation for two-bridge knots.
ε e
-bn m
an -b m2
a2
t
2a1 2b1
2bn
2an
2b1
2a1 A
e
2a1
b g
tea1
m1 -b
t
Figure 8. Standard representation of two-bridge knots.
Now we show thatte¼t1l tmtlt1m (note that no disk bounded byeand properly embedded in H is disjoint from A). Referring to Figure 9, the following ‘‘lantern’’
relationtg2td1td2¼tetbtz holds (see [23]). So we obtainz¼tatgt1b t1a ðgÞand therefore tz¼tatgt1b t1a tgtatbtg1t1a . Sincetd1¼td2¼1 we havete¼tg2t1z t1b ¼tg2tatgt1b t1a tg1 tatbtg1t1a t1b . Now, using the relations of (1) we get
te¼tg2tatgtb1t1a tg1tatbtg1t1a t1b ¼tgtatgtat1b t1a tg1tatbtg1t1a t1b
¼tgtatgt1b t1a tbtg1tatbtg1t1a t1b ¼tgtat1b tgt1a tg1tbtatbtg1t1a t1b
¼tgtat1b t1a tg1tatbtatbtg1t1a t1b ¼tgt1b t1a tbtg1tatbtatbtg1t1a t1b
¼tgt1b t1a tbtg1tatatbtatg1t1a t1b ¼tgt1b t1a tbtg1tatatbtg1t1a tgt1b
¼t1m t1a tmtatatmt1a t1m ¼t1m t1a tmtatmt1m tatmt1a t1m
¼t1h tatmtht1a t1m ¼t1l tmtlt1m : r
For example, the figure-eight knotbð5=2Þ, which has Conway parameters½2;2, is the knotKc withc¼tbtatbt1m te (see Figure 10).
Acknowledgements. The authors would like to thank Sylvain Gervais and Andrei Vesnin for their helpful suggestions. We also would like to thank the Referee for his valuable comments and remarks. Work performed under the auspices of the G.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
d1
e b d2 z
A Figure 9.
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Received 9 September, 2002; revised 12 December, 2002
A. Cattabriga, Department of Mathematics, University of Bologna, Italy Email: [email protected]
M. Mulazzani, Department of Mathematics and C.I.R.A.M., University of Bologna, Italy Email: [email protected]
