(1,1)-knots via the mapping class group of the twice ... · adv. geom. 4 (2004), 263–277 advances...
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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
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Received 9 September, 2002; revised 12 December, 2002
A. Cattabriga, Department of Mathematics, University of Bologna, ItalyEmail: [email protected]
M. Mulazzani, Department of Mathematics and C.I.R.A.M., University of Bologna, ItalyEmail: [email protected]
ð1; 1Þ-knots via the mapping class group of the twice punctured torus 277