Adv. Geom. 4 (2004), 263–277
Advances in Geometry
( 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 MCG2 ðTÞ. We prove that every ð1; 1Þ-knot in a lens space
Lð p; qÞ can be represented by the composition of an element of a certain rank two free subgroup of MCG2 ðTÞ with 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 type ðk; ck þ 2Þ.
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 of ð1; 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 arcs ft1 ; . . . ; tb g properly embedded in a
handlebody H is trivial if there exist b mutually disjoint discs D1 ; . . . ; Db H H such
that ti V Di ¼ ti V qDi ¼ ti , ti V Dj ¼ q and qDi ti H qH for all i; j ¼ 1; . . . ; b and
i 0 j. Let M ¼ H Uj H 0 be a genus g Heegaard splitting of a closed orientable 3manifold M and let F ¼ qH ¼ qH 0 ; a link L H M is said to be in b-bridge position
with respect to F if: (i) L intersects F transversally and (ii) L V H and L V H 0 are both
264
Alessia Cattabriga and Michele Mulazzani
H
A
ϕ
H'
A'
Figure 1. A ð1; 1Þ-decomposition.
the union of b mutually 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; bÞdecomposition. Note that a ð0; bÞ-link is a link in S 3 which admits a b-bridge presentation in the usual sense. So the notion of ðg; bÞ-decomposition of links in 3manifolds generalizes the classical bridge (or plat) decomposition of links in S 3 (see
[7]). Obviously, a ðg; 1Þ-link is a knot, for every g d 0.
Therefore, a ð1; 1Þ-knot K is a knot in a lens space Lð p; qÞ (possibly in S 3 ) which
admits a ð1; 1Þ-decomposition
ðLðp; qÞ; KÞ ¼ ðH; AÞ Uj ðH 0 ; A 0 Þ;
where j : ðqH 0 ; qA 0 Þ ! ðqH; qAÞ is an (attaching) homeomorphism which reverses
the standard orientation on the tori (see Figure 1). It is well known that the family
of ð1; 1Þ-knots contains all torus knots (trivially) and all two-bridge knots (see [16])
in S 3 .
In this paper we develop an algebraic representation of ð1; 1Þ-knots through elements of MCG2 ðTÞ, the mapping class group of the twice punctured torus. In Section 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
MCG2 ðTÞ which is the composition of an element of a certain rank two free subgroup and of a standard element only depending on the ambient space Lðp; 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; 1Þknots in S 3 : 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 type ðk; ck þ 2Þ, with c; k > 0 and k
odd.
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
265
b
T
α
P1
g
P2
Figure 2. Generators of MCG2 ðTÞ.
In what follows, the symbol Lðp; qÞ will denote any lens space, including S 3 ¼
Lð1; 0Þ and S 1 S 2 ¼ Lð0; 1Þ. Moreover, homotopy and homology classes will be
denoted 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 ; . . . ; Pn g be a finite
set of distinguished points of Fg , called punctures. We denote by HðFg ; PÞ the group
of orientation-preserving homeomorphisms h : Fg ! Fg such that hðPÞ ¼ P. The
punctured mapping class group of Fg relative to P is the group of the isotopy classes
of elements of HðFg ; PÞ. Up to isomorphism, the punctured mapping class group
of 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 by
MCGn ðFg Þ. Moreover, for isotopy classes we will use the same symbol of the representing homeomorphisms.
The n-punctured pure mapping class group of Fg is the subgroup PMCGn ðFg Þ of
MCGn ðFg Þ consisting of the elements pointwise fixing the punctures. There is a
standard exact sequence
1 ! PMCGn ðFg Þ ! MCGn ðFg Þ ! 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 MCG2 ðTÞ. According to previously cited papers, a set of generators for
MCG2 ðTÞ is given by a rotation r of p radians which exchanges the punctures and
the right-handed Dehn twists ta ; tb ; tg around the curves a; b; g respectively, as depicted in Figure 2. Since r commutes with the other generators, we have
MCG2 ðTÞ G PMCG2 ðTÞ l Z2 :
266
Alessia Cattabriga and Michele Mulazzani
T
T
m
g
T
b
h
α
T
l
Figure 3. Action of tm and tl .
The following presentation for PMCG2 ðTÞ has been obtained in [22]:
hta ; tb ; tg j ta tb ta ¼ tb ta tb ; ta tg ta ¼ tg ta tg ; tb tg ¼ tg tb ; ðta tb tg Þ 4 ¼ 1i:
ð1Þ
The group PMCG2 ðTÞ (as well as MCG2 ðTÞ) naturally maps by an epimorphism
to the mapping class group of the torus MCGðTÞ G SLð2; ZÞ, which is generated by
ta and tb ¼ tg . So we have an epimorphism
W : PMCG2 ðTÞ ! SLð2; ZÞ
1 0
1 1
defined by Wðta Þ ¼
and Wðtb Þ ¼ Wðtg Þ ¼
.
1 1
0 1
The group ker W will play a fundamental role in our discussion. In order to investigate its structure, let us consider the two elements tm ¼ tb tg1 and tl ¼ th t1
a , where
th is the right-handed Dehn twist around the curve h depicted in Figure 3. The e¤ect
of tm and tl is to slide one puncture (say P2 ) respectively along a meridian and along
a longitude of the torus, as shown in Figure 3. Observe that, since h ¼ t1
m ðaÞ, we
have th ¼ t1
m t a tm .
The following result can be obtained from [3, Theorem 1] and [2, Theorem 5] by
classical techniques.
Proposition 1. The group ker W is freely generated by tm ¼ tb tg1 and tl ¼ th t1
a , where
th ¼ t1
m t a tm .
Now, let K H Lðp; qÞ be a ð1; 1Þ-knot with ð1; 1Þ-decomposition ðLðp; qÞ; KÞ ¼
ðH; AÞ Uj ðH 0 ; A 0 Þ and let m : ðH; AÞ ! ðH 0 ; A 0 Þ be a fixed orientation-reversing
homeomorphism, then c ¼ jmjqH is an orientation-preserving homeomorphism of
ðqH; qAÞ ¼ ðT; fP1 ; P2 gÞ. Moreover, since two isotopic attaching homeomorphisms
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
267
produce equivalent ð1; 1Þ-knots, we have a natural surjective map from the twice
punctured mapping class group of the torus MCG2 ðTÞ to the class K1; 1 of all ð1; 1Þknots
Y : c A MCG2 ðTÞ 7! Kc A K1; 1 :
q s
If WðcÞ ¼
, then Kc is a ð1; 1Þ-knot in the lens space Lðj pj; jqjÞ [4, p. 186],
p r
and therefore it is a knot in S 3 if and only if p ¼ G1.
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 S 1 S 2 ;
ii) if c ¼ ta , then Kc is the trivial knot in S 3 .
Moreover, it is possible to prove that if c ¼ ta tb ta ta tg ta , then Kc is the knot
S 1 fPg H S 1 S 2 , where P is any point of S 2 . So, in this case, Kc is a standard
generator for the first homology group of S 1 S 2 .
Every element c of MCG2 ðTÞ can be written as c ¼ c 0 r k , k A f0; 1g, where
0
c A PMCG2 ðTÞ. 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 to
consider the restriction
Y 0 ¼ YjPMCG2 ðTÞ : c A PMCG2 ðTÞ 7! Kc A K1; 1 :
3 Standard decomposition
In this section we show that every ð1; 1Þ-knot K H Lðp; qÞ admits a representation
by the composition of an element in ker W and 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 MCG2 ðTÞ, has been obtained in [6, Theorem 3].
First of all, we deal with trivial knots in lens spaces. Let T be the subgroup of
PMCG2 ðTÞ generated by ta and tb . There exists a disk D H H, with A V D ¼
A V qD ¼ A and qD A H T, such that D V a ¼ D V b ¼ q. So any element of T
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 of T, as will be proved in
Proposition 3.
We need a preparatory result.
Lemma 2. Let K be a ð1; 1Þ-knot in Lð p; qÞ. Then, for
q
qr ps ¼ 1 there exists c A PMCG2 ðTÞ, with WðcÞ ¼
p
each
r; s A Z such that
s
, such that K ¼ Kc .
r
268
Alessia Cattabriga and Michele Mulazzani
q s
Proof. Let K ¼ Kc , with WðcÞ ¼
. Since qr ps ¼ 1, there exist c A Z such
p r
that r ¼ r þ cp and s ¼ s þ cq. If c ¼ ctbc , we have Kc ¼ Kc , since tbc can be extended toa homeomorphism
of the pair ðH; AÞ. Moreover WðcÞ ¼ WðcÞWðtbc Þ ¼
1 c
q s
q s þ cq
¼
.
r
0 1
p r
p r þ cp
For integers p; q such that 0 < q < p and gcdðp; qÞ ¼ 1 consider the sequence of
equations of the Euclidean algorithm (with r0 ¼ p, r1 ¼ q):
r0 ¼ a 1 r1 þ r2
r1 ¼ a2 r2 þ r3
..
.
rm2 ¼ am1 rm1 þ rm
rm1 ¼ am rm ;
with r1 > r2 > > rm1 > rm ¼ 1.
The ai ’s are the coe‰cients of the continued fraction
p
1
¼ a1 þ
q
a2 þ
1
a3 þþa1m
:
In the following we will use the notation p=q ¼ ½a1 ; a2 ; . . . ; am .
Proposition 3. The trivial knot in S 3 ¼ Lð1; 0Þ is represented by c1; 0 ¼ tb ta tb .
The trivial knot in S 1 S 2 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 Lð p; qÞ is represented by
( a
taa1 tb 2 . . . taam
if m is odd;
cp; q ¼ a1 a2
m
ta tb . . . ta
t
t
t
if m is even:
b a b
b
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:
8
> 1
>
>
< a
q s
1
¼ >
1
p r
>
>
:
a1
1 a2
1 0
0
...
1
0 1
am 1
1 a2
0
0
1 am
...
1
0 1
0 1
1
if m is odd;
1
0
if m is even.
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
1
Since
¼
ai
ment is obtained.
Wðtaai Þ
0
1
ai
, Wðtb Þ ¼
1
0
269
ai
0 1
, and Wðtb ta tb Þ ¼
, the state1
1 0
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 ker W such that
K ¼ Kc , with c ¼ c 0 cp; q ¼ cp; q c 00 .
Proof. By Lemma 2, there exists c, with WðcÞ ¼ Wðcp; q Þ, such that K ¼ Kc . It suf00
1
fices to define c 0 ¼ cc1
r
p; q and c ¼ cp; q c.
A representation c A PMCG2 ðTÞ of a ð1; 1Þ-knot will be called standard if c is 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 S 1 S 2 , for all
c A Z.
4 Representation of torus knots
In this section we give a standard representation for all torus knots in S 3 . Let
K ¼ tðk; hÞ be a torus knot of type ðk; hÞ. Then gcdðk; hÞ ¼ 1, and we can assume
that K lies on the boundary T ¼ qH of a genus one handlebody H canonically embedded in S 3 . The homology class of K is hl þ km, where l and m respectively denote
a longitude and a meridian of T. By slightly pushing (the interior of ) an arc A 0 H K
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; hÞ H S 3 is the ð1; 1Þ-knot Kc with:
c¼
h
Y
ðtbði1Þk=hcbik=hc
t1
m
l Þtb ta tb ;
i¼1
1
where tm ¼ tb tg1 and tl ¼ t1
m t a tm t a .
Proof. Up to isotopy, we can suppose that the arc A ¼ Kc intðA 0 Þ lies on qH, as
in Figure 4. The arc A can be transformed into an arc A~ in such a way that A~ U A 0 is
a trivial knot in S 3 , represented by the standard homeomorphism c1; 0 ¼ tb ta tb , via a
suitable sequence of homeomorphisms tl and tm , according to the following algorithm. Consider the sequence of equations:
270
Alessia Cattabriga and Michele Mulazzani
H
A
A'
Figure 4.
l
l m
Figure 5.
k ¼ q1 h þ r1 ;
2k ¼ q2 h þ r2 ;
..
.
hk ¼ qh h þ rh ;
where 0 c 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 tl tmqi qi1 , for i ¼ 1; . . . ; h. Figure 5
depicts the e¤ect of tl and tl tm on A. As a consequence, the homeomorphism f ¼
ch ch1 . . . c1 transforms the arc A into the arc A~ (Figure 6 shows the case tð5; 7Þ),
and therefore we have c1; 0 ¼ fc. So f1 c1; 0 represents the torus knot tðk; hÞ.
r
1 1 2 1 1 1 3
For example, tð5; 7Þ ¼ Kc , with c ¼ t1
l ðtm tl Þ tl ðtm tl Þ tb ta tb (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.
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
271
l
l m
l m
l
l m
l m
l m
Figure 6. Trivialization of tð5; 7Þ.
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 Gn ðwÞ,
where w is equal to
ðk3Þ=2
Y cðk1Þ=2
Y
i¼0
x1iðckþ2Þþjk
cðkþ1Þ=2
Y
j¼0
(subscripts are taken modulo n).
l¼0
x1
ckðk1Þ=2iðckþ2Þlk
cðk1Þ=2
Y
m¼0
x1ðk1Þðckþ2Þ=2þmk
272
Alessia Cattabriga and Michele Mulazzani
Proof. Let r ¼ ðk 1Þ=2. From Theorem 5 we have tðk; ck þ 2Þ ¼ Kc with
1 r 1 c 1 r c 1 1
c ¼ ðtc
l tm Þ tl ðtl tm Þ tl tm tl tb ta tb . Applying [5, Proposition 1], we obtain
3
p1 ðS tðk; ck þ 2ÞÞ ¼ ha; g j rða; gÞi, with rða; gÞ ¼ ðg1 a crþ1 g1 acðrþ1Þ1 Þ r g1 a crþ1 .
Then H1 ðS 3 tðk; ck þ 2ÞÞ ¼ ha; g j a kgi. Since, up to equivalence, of ðgÞ ¼ 1, we
have of ðaÞ ¼ k. We set a ¼ xg k , therefore p1 ðS 3 tðk; ck þ 2ÞÞ ¼ hx; g j rðx; gÞi,
with rðx; gÞ ¼ ðg1 ðxg k Þ 1þcðk1Þ=2 g1 ðgk x1 Þ 1þcðkþ1Þ=2 Þðk1Þ=2 g1 ðxg k Þ 1þcðk1Þ=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; 7Þ admits the cyclic presentation Gn ðwÞ, where
1 1 1
1 1 1 1
w ¼ x15 x20 x25 x1
24 x19 x14 x9 x8 x13 x18 x17 x12 x7 x2 x1 x6 x11 :
5 Representation of two-bridge knots
In this section we give a standard representation for all two-bridge knots in S 3 . Let
bða=bÞ be a non-trivial two-bridge knot in S 3 of type ða; bÞ. Then we can assume
gcdða; 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 number
of even parameters ½2a1 ; 2b1 ; . . . ; 2an ; 2bn (see Figure 7), satisfying the following relation:
a
1
¼ 2a1 þ
b
2b1 þ
1
2a2 þþ2b1
:
n
Theorem 7. The two-bridge knot bða=bÞ H S 3 having Conway parameters ½2a1 ; 2b1 ; . . . ;
2an ; 2bn is the ð1; 1Þ-knot Kc with:
b1 a1
n an
c ¼ tb ta tb tb
m t e . . . tm t e ;
1
where te ¼ t1
l tm tl tm is the right-handed Dehn twist around the curve e depicted in
Figure 8.
b1 a1
n an
Proof. Figure 8 shows the result of the application of tb
m te . . . tm te . By applying
c1; 0 ¼ tb ta tb we obtain the two-bridge knot with Conway parameters ½2a1 ; 2b1 ; . . . ;
2an ; 2bn .
2b1
2a1
2bn
2an
Figure 7. Conway presentation for two-bridge knots.
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
273
e
g
b
a
te 1
2a1
A
-b1
m
2bn
2a n
-bn an
t
m e
-b 2 a2
t
m ε
2b1
2b1
2a1
2a1
Figure 8. Standard representation of two-bridge knots.
274
Alessia Cattabriga and Michele Mulazzani
T
e
b
d1
g
d2
z
g
A
Figure 9.
1
Now we show that te ¼ t1
l tm tl tm (note that no disk bounded by e and properly
embedded in H is disjoint from A). Referring to Figure 9, the following ‘‘lantern’’
1
relation tg2 td1 td2 ¼ te tb tz holds (see [23]). So we obtain z ¼ ta tg t1
b ta ðgÞ and therefore
1 1
1 1
2 1 1
1 1
tz ¼ ta tg tb ta tg ta tb tg ta . Since td1 ¼ td2 ¼ 1 we have te ¼ tg tz tb ¼ tg2 ta tg t1
b ta tg 1 1 1
ta tb tg ta tb . Now, using the relations of (1) we get
1
1 1 1
1 1 1
1 1 1
te ¼ tg2 ta tg tb1 t1
a tg ta tb tg ta tb ¼ tg ta tg ta tb ta tg ta tb tg ta tb
1
1
1 1 1
1
1 1
1 1 1
¼ tg ta tg t1
b ta tb tg ta tb tg ta tb ¼ tg ta tb tg ta tg tb ta tb tg ta tb
1 1
1 1 1
1 1
1
1 1 1
¼ tg ta t1
b ta tg ta tb ta tb tg ta tb ¼ tg tb ta tb tg ta tb ta tb tg ta tb
1
1
1 1 1
1 1
1
1 1
1
¼ tg t1
b ta tb tg ta ta tb ta tg ta tb ¼ tg tb ta tb tg ta ta tb tg ta tg tb
1
1 1
1 1
1
1 1
¼ t1
m ta tm ta ta tm ta tm ¼ tm ta tm ta tm tm ta tm ta tm
1 1
1
1
¼ t1
h ta tm th ta tm ¼ tl tm tl tm :
r
For example, the figure-eight knot bð5=2Þ, which has Conway parameters ½2; 2, is the
knot Kc with c ¼ tb ta tb t1
m 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 selected research topics.
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
275
-1
m
te
Figure 10. Standard representation of the figure-eight knot.
References
[1] J. Berge, The knots in D 2 S 1 which have nontrivial Dehn surgeries that yield D 2 S 1 .
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 #2764
Zbl 0157.30904
276
Alessia Cattabriga and Michele Mulazzani
[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 Ramifications 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. Topology 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 discriminant set of a stable map. Pacific J. Math. 195 (2000), 101–156. MR 2001i:57026
Zbl 01537853
[17] C. Labruère, L. Paris, Presentations for the punctured mapping class groups in terms of
Artin 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 have
the 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; 1Þknots. 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’’ (Bielefeld, 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
ð1; 1Þ-knots via the mapping class group of the twice punctured torus
277
[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, Italy
Email: cattabri@dm.unibo.it
M. Mulazzani, Department of Mathematics and C.I.R.A.M., University of Bologna, Italy
Email: mulazza@dm.unibo.it