TOPICS ON D E H N S U R G E R Y
By
Xingru Zhang
B.Sc. of Mathematics, Nanjing Institute of Posts and Telecommunications
A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF
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T H E F A C U L T Y OF G R A D U A T E STUDIES
MATHEMATICS
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T H E UNIVERSITY OF BRITISH COLUMBIA
January 1991
© Xingru Zhang,
1991
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Department
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M.flfjt.l^ri(3t f
The University of British Columbia
Vancouver, Canada
Date
DE-6 (2788)
A^-.|
tf?i
Abstract
Cyclic surgery on satellite knots i n S
3
knot i n S
3
is classified and a necessary condition is given for a
to admit a nontrivial cyclic surgery with slope m/l, \m\ > 1. A complete classi-
fication of cyclic group actions on the Poincare sphere with 1-dimensional fixed point sets is
obtained. It is proved that the following knots have property I, i.e. the fundamental group
of the manifold obtained by Dehn surgery on such a knot cannot be the binary icosahedral
group I120, the fundamental group of the Poincare homology 3-sphere: nontrefoil torus knots,
satellite knots, nontrefoil generalized double knots, periodic knots with some possible specific
exceptions, amphicheiral strongly invertible knots, certain families of pretzel knots.
Further
the Poincare sphere cannot be obtained by Dehn surgery on slice knots and a certain family
of knots formed by band-connect sums. It is proved that if a nonsufficiently large hyperbolic
knot i n S
3
admits two nontrivial cychc Dehn surgeries then there is at least one nonintegral
boundary slope for the knot. There are examples of such knots. Thus nonintegral boundary
slopes exist.
ii
Table of Contents
Abstract
"
List of Figures
v
Acknowledgements
V
1
Introduction
v
"
1
2
O n C y c l i c Surgery
1
1.1
Introduction
1
1.2
Preliminaries
5
1.2.1
C M . Gordon's Lemma
5
1.2.2
D. Gabai's Results
6
1.3
Proof of Theorem 1.1.4
9
1.4
Proof of Proposition 1.1.1
11
1.5
Examples, Remarks and Open Problems
12
On Property I
18
2.1
Introduction
18
2.2
Prehminaries
20
2.2.1
20
The Casson Invariant and Property I (I)
iii
2.2.2
2.2.3
The Conway Polynomial and the Kauffman Bracket Polynomial
25
26
2.3
Cyclic Actions on the Poincare Homology 3-Sphere
28
2.4
Knots Having Property I or I
35
2.4.1
35
2.4.2
2.5
3
The Rohlin Invariant and the A r f Invariant
Torus knots, Slice Knots and Knots Formed by Band Connect Sums . . .
Satellite Knots and Generalized Double Knots
40
2.4.3
Periodic Knots
43
2.4.4
Strongly Invertible Knots
46
2.4.5
Pretzel Knots
48
2.4.6
Knots up to 9 Crossings
50
Concluding Remarks and Open Problems
O n Boundary Slopes
52
55
3.1
Introduction
55
3.2
Proof of Theorem 3.1.1
56
3.3
Proof of Lemma 3.1.1
58
3.4
Properties of <p(K) and Open Problems
62
Bibliography
64
iv
List of Figures
1.1
Fintushel-Stern knots K
13
1.2
Berge-Gabai knots J
14
2.3
several surgery presentations of the Poincare sphere
29
2.4
a band-connect sum of two knots
36
2.5
r-moves
38
2.6
Ki# K
2.7
a generalized double knot
2.8
generalized twisted knot K
2.9
8i8 has 4i as a factor knot
2.10
a pretzel knot of type K(pi, • • • ,p )
2.11
a pretzel knot of type (2m -f 1,2m + 1,2m + 1) and its factor knot
51
2.12
a Montesinos knot of type (px/gi, „ . , p / o )
53
3.13
surgery on (—2,3, 7) pretzel knot and double branched covering
59
n
b
2
n
is r-equivalent to K ^K
1
39
2
41
42
VA
45
49
m
n
n
3.14 branched sets of 18- 19-surgeries on the (—2,3,7) pretzel knot
v
61
Acknowledgements
I wish to express m y gratitude to m y supervisor, Professor E r h a r d L u f t , for his invaluable
guidance, encouragement and support. I also would hke to thank the University of British
Columbia for its generous financial assistance. F i n a l thanks go to m y family, especially to my
wife, Lijuan Zhang, for their emotional support.
vi
Introduction
One of the basic methods to construct closed orientable 3-manifolds is by Dehn surgery on
knots or links i n the 3-sphere S ,
3
which was introduced by M . Dehn i n 1910 [18].
It is the
process of removing a regular neighborhood of the knot or hnk and sewing it back i n via a
homeomorphism on the boundary torus or tori respectively of the regular neighborhood. The
fact that every closed orientable 3-manifold can be obtained by Dehn surgery on a link in S
3
was proven by A.Wallace [80] and W . B . B . . Lickorish [49] i n the early sixties.
Thus a good
understanding of Dehn surgery is important for the theory of 3-manifolds. However, even in
the case of knots i n 5 , it is i n general not known which manifold can be obtained by which
3
surgery on which knot. There are very few classes of knots on which the manifolds obtained
by Dehn surgery are explicitly known (among them are the torus knots [56]). Around the late
seventies a general picture of 3-manifolds obtained by surgery on links was described by W .
Thurston through his geometric approach [78] [77]. In particular he proved that if a knot in
S
3
is neither a satellite knot nor a torus knot then the interior of the knot complement admits
a complete hyperbolic structure of finite volume (such a knot is called a hyperbolic knot) and
Dehn surgeries on a hyperbolic knot yield hyperbolic manifolds except for finitely many cases.
It is also well known that if the complement of a hyperbolic knot contains no incompressible
nonboundary parallel closed surfaces, then again except for finitely many cases Dehn surgeries
on the knot yield hyperbolic manifolds that do not contain incompressible closed surfaces. For
a satellite knot, nonboundary parallel incompressible tori i n the knot complement will remain
incompressible i n manifolds obtained by Dehn surgery on the satellite knot except for finitely
many cases, unless the knot is a cabled knot [16]. Naturally questions about those exceptional
surgeries i n the sense described above are of considerable interest. In this paper we address
three topics concerning Dehn surgery along this line.
vii
Topic 1. W h i c h Dehn surgery on which knot in S
which Dehn surgery on which knot in S
3
3
can yield a lens space? More generally
can yield a manifold with cychc fundamental group?
Topic 2. W h i c h Dehn surgery on which knot i n 5
3
can yield the Poincare homology 3-
sphere? More generally which Dehn surgery on which knot i n S
can yield a manifold with
3
fundamental group I\2Q, the binary icosahedral group?
Topic 3. Axe there nonintegral boundary slopes for knots i n 5 ?
3
The main results of the thesis are the following.
On Topic 1: Cychc surgery on satellite knots i n S is classified and a necessary condition is
3
given for a knot i n S
3
to admit a nontrivial cychc surgery with slope m/Z, |m| > 1. A theorem
of Gabai is proved by using the /3-norm based sutured 3-manifold theory of M . Scharlemann.
On Topic 2: A complete classification of cychc group actions on the Poincare sphere with
1-dimensional fixed point sets is obtained. It is proved that the fundamental group of a manifold
obtained by Dehn surgery on the following knots cannot be the binary icosahedral group IHQ:
nontrefoil torus knots, satellite knots, nontrefoil generalized double knots, periodic knots with
some possible specific exceptions, amphicheiral strongly invertible knots, certain families of
pretzel knots. The Poincare sphere cannot be obtained by Dehn surgery on slice knots and a
certain family of knots formed by band-connect sums.
On Topic 3: It is proved that if a hyperbolic knot i n S
3
admits two nontrivial cychc surgeries
then there exists at least one nonintegral boundary slope. There are examples of such knots.
Thus nonintegral boundary slopes exist.
viii
Chapter 1
On Cyclic Surgery
1.1
Introduction
We work in all three chapters in the P L category. A P L homeomorphism we simply call an
isomorphism. Our reference for basic terminology is [37] and [65].
We first describe (Dehn) surgery.
This operation can be done along any knot K in any
orientable 3-manifold M. Namely, remove a tubular neighborhood N(K)
back in by an isomorphism of tori. Let E = M — intN(K)
of K in M and sew it
and choose two simple closed curves,
fi and A, on dE such that H\(dE) = Z[p] + Z[X]. Then the different surgeries (sewings) can
be parametrized by so called surgery slopes mfl 6 Q U {1/0}
(m,l)
where m and / are integers with
= 1; namely corresponding to the surgery with slope m/l the simple closed curve (up
to isotopy of torus) on dE with homology class m[p] + l[X] in H\(dE) = Z[p] + Z[X] bounds
a meridian disc in the sewn solid torus.
Such a pair of curves p. and A is called a framing
pair. We denote the resulting manifold by M(K,m/l).
If Af is a homology 3-sphere (i.e.
a
3-manifold with the same homology as the 3-sphere), then p and A in dE can be chosen to he
a preferred meridian-longitude framing pair so that [p.] — 0 in Hi(N(K))
in H\(E)
= Z[X] and [A] = 0
— Z[p]. Unless otherwise specified all surgeries on knots in homology 3-spheres are
performed with respect to a preferred meridian-longitude framing pair. Note that if K is a knot
in a homology 3-sphere then Hi(M(K,m/l))
= Z\ \- Hence M(K,m/l)
m
is a homology 3-sphere
iff |m| = 1.
Let S (K,
3
m/l) denote the closed orientable 3-manifold obtained by surgery with slope m/l
1
Chapter 1. On Cyclic Surgery
along a knot K in S . If S (K,m/l)
3
3
2
is a manifold with cychc fundamental group (if so the
group is Z| |), then the corresponding m//-surgery is called a cyclic surgery or a Z| | surgery.
m
m
In particular if S (K, m/l) is a lens space, then the corresponding m//-surgery is also called a
3
lens space surgery. It is not known whether or not lens spaces are the only closed orientable
3-manifolds with cychc fundamental groups. We call a closed orientable 3-manifold a fake lens
space if the manifold has cychc fundamental group but is not homeomorphic to a lens space.
Let O denote the trivial knot in 5 then surgeries on O produce all lens spaces (including
3
S and S x S ) and S (0, m/l) = l ( m , /).
3
2
1
3
In [56] L. Moser classified all manifolds obtained by surgery on torus knots. In particular
she proved the following (see also [39] Chapter IV)
Theorem 1.1.1 ([56]) Nontrivial surgery with slope m/n on a nontrivial torus knot T(p,q)
gives a manifold with cyclic fundamental group iff m = npg ± 1 and the manifold obtained
the lens space L(m,nq ).
2
J. Bailey and D. Rolfsen [2] gave the first example of surgery on a nontorus knot that
produces a lens space. They showed that —23 surgery on the (ll,2)-cable on the trefoil knot
gives the lens space L(23,7). Later R. Fintushel and R. Stern [21] constructed lens spaces by
surgery on a variety of nontorus knots (see also [54]). In particular they proved the following
(see also [28] Theorem 7.5 )
Theorem 1.1.2 ([21]) Nontrivial surgery with slope m/n on a nontrivial cabled knot C(r,s
on a nontrivial torus knot T(p, q) gives a manifold with cyclic fundamental group iffs = 2, r
2pq ± 1, m/n = Apq ± 1 and the manifold is the lens space L(4pq ± l,4g ).
2
Major progress on cychc surgery was made in M . Culler, C M . Gordon, J. Luecke and P.B.
Shalen's paper [16]. They showed, in particular, the following
Chapter 1.
On Cyclic Surgery
3
Theorem 1.1.3 ([16]) If a nontorus knot in S admits a cyclic surgery, then the surgery slope
3
is an integer. Any nontorus knot admits at most two nontrivial cyclic surgeries and if that is
the case, then the two slopes are successive integers.
Our first result of this chapter gives a complete classification of cychc surgery on satellite
knots, that is
Theorem 1.1.4 Nontrivial surgery with slope m/n on a satellite knot K in S gives a manifold
3
with cyclic fundamental group iff K is a knot as in Theorem 1.1.2, i.e. a cabled knot C(r,s) on
a torus knot T(p, q) with s = 2, r = 2pq ± 1, m/n = 4pq ± 1 and the manifold is the lens space
L(4pq±l,4q ).
2
Theorem 1.1.4 was obtained in the author's paper [83] and was also independently obtained
by Y . Wu [82] and by S. Bleiler and R . A . Litherland [10].
Corollary 1.1.1 ([22]) Satellite knots in S have property P.
3
Therefore classification of cychc surgery on knots in S reduces to hyperbolic knots. There
3
do exist hyperbolic knots admitting nontrivial cychc surgery and infinitely many such examples
can be found in [21] and [25] (see Examples 1.5.1-2). S. Wang and Q. Zhou showed in [81] that
if a nontorus knot in S admits a symmetry (i.e. is invariant under a finite group action on
3
S ) which is not a strong inversion (see section 2.4.4 for the definition), then there exists no
3
nontrivial cychc surgery on the knot. They also showed that no surgery on a symmetric knot
can produce a fake lens space. M . Takahashi showed in [74] that any nontorus 2-bridge knot
does not admit a cychc surgery. As D . Gabai [24] has given a positive answer to the Poenaru
conjecture, 0-surgery on any nontrivial knot in S never give a manifold with infinite cychc
3
fundamental group. Hence if If is a hyperbolic knot which admits a nontrivial cychc surgery
then the surgery slope is an integer m with 0 < |m| < oo.
4
Chapter 1. On Cyclic Surgery
Our second result in this chapter gives a necessary condition for a knot in S to admit a
3
nontrivial cyclic surgery with slope m/l, |m| > 1.
Proposition 1.1.1
Let M be a homology 3-sphere, let K C M be a knot and let
M(K,m/l)
be the manifold obtained by surgery on M along K with slope m/l, \m\ > 1. Let p : M /i
m
—•
M(K, m/l) be the \m\-fold cyclic unbranched regular covering defined by ker(ir(M(K, m/l)) —•
Hi(M(K, m/l)) = Z ) and let q : M(m) —• M be the \m\-fold cyclic branched regular covering
m
of M with branch set K in M.
Then M /i
m
is a homology 3-sphere iff M(m) is a homology
3-sphere.
Corollary 1.1.2
If a knot K in S
admits a nontrivial cyclic surgery with slope m/l, \m\ > 1
3
then the \m\-fold cyclic branched cover of S
lni=i ' Ax(e
m
2,r,J
'/l l)| = 1 where A (t)
m
K
3
branched over K is a homology 3-sphere and t
is the Alexander polynomial of K (see [19]). •
It was shown by S. Bleiler and R. Litherland in [7] that the projective space RP cannot be
3
obtained by surgery on any nontrivial symmetric knot in S . This result has been generalized
3
by S. Wang and Q. Zhou in [81] to : No nontrivial symmetric knot in S admits /^-surgery.
3
As a special case of Corollary 1.1.2 we have that if some knot K in S admits Z2-surgery, then
3
A K ( - 1 ) ) the determinant of ii!", is 1 or -1. This criterion is quite effective; in fact among all
249 nontrivial knots of 10 or less crossings only two of them, IO124 and IO153, have determinants
± 1 . But these two knots are symmetric [12]
Corollary 1.1.3
(IO124
is the (3,5) torus knot), hence we have
Surgery on any nontrivial knot of 10 or less crossings cannot give a manifold
M with ir (M) = Z . •
x
2
The rest of this chapter is organized as follows. In the next section, we recall some known
results about surgery on knots in a solid torus, which are needed to prove Theorem 1.1.4. A
theorem of D. Gabai is reproved by using M. Scharlemann's /3-norm based sutured 3-manifold
theory. In section 1.3 we give a proof of Theorem 1.1.4 which is basically a delicate consequence
Chapter 1. On Cyclic Surgery
5
of Theorems 1.1.1-3, Lemma 1.2.1 and Theorems 1.2.1-2. A proof of Proposition 1.1.1 is given
in section 1.4. Section 1.5 consists of examples, remarks and open problems.
1.2
Preliminaries
Since a satellite knot is contained in a nontrivial solid torus of S nontrivially (i.e. not isotopic
3
to the core of the solid torus and is not contained in a 3-ball of the solid torus), one may
obtain some information by first considering surgery on the solid torus along the knot. Explicit
homological information about surgery on knots in a solid torus was given by C M . Gordon in
[28], which are to be recorded in section 1.2.1 (Lemma 1.2.1). D. Gabai proved fundamental
theorems concerning surgery on knots in a solid torus [22] [25]. In section 1.2.2 two main results
from [22] and [25], Theorems 1.2.1-2 below, are introduced.
1.2.1
G . M . Gordon's Lemma
Let K C S be a satellite knot and let K* be a nontrival companion knot of K. Let N* = K* x
3
D C S be a solid torus neighborhood of K* in S with K C intN* and let E" = S - intN*.
2
3
3
3
Let p*, A* be apreferred meridian-longitude pair of ON* = dE*, that is, Hi(dN*) = Hi(dE') =
Z[p*] + Z[X*], p* = 0 in H^N*) = Z[X*] and [A*] = 0 in H^E*) = Z[p*}.
Suppose [A'] = u>[\*] in H\(N*). We may assume that u > 0 by choosing a proper orientation for K. Then w > 0 is the winding number of K in N*.
Let N = K x D C tniiV* be a solid torus neighborhood of K in N* and let E = S - intN
2
3
and EQ — N* — intN. Let p, A be a preferred meridian-longitude pair of dN = dE, that is,
HiidN) = H\(dE) = Z[p] + Z[X], u = 0 in Hi(N) = Z[X] and [A] = 0 in H (E) = Z[u]. Then
X
tfi^o) = Z\p](BZ[\*] [A] = u[X*] in H^EQ) and [fi*] = u[p] in # i ( £ ) (by choosing proper
t
0
orientation for p and A).
Let N*(K,m/l) denote the manifold obtained by surgery along K in N*. As S (K,m/l) =
3
6
Chapter 1. On Cyclic Surgery
E*UN*(K, m/l), we may obtain some information about S (K, m/l) by first considering surgery
3
in N* along K. The following lemma proved in [28] by C M . Gordon gives precise homological
information about
N*(K,m/l).
Lemma 1.2.1 ([28]) Lemma 3.3) (i). Hi(N (K,m/l))*
m
(ii). ker(Hi(6N*(K, m/l)) —> HriN^K,
Z© Z
.
(WyTn)
m/l))) is the cyclic subgroup ofHi(dN'{K,
m/l))
generated by
[/*•],
1.2.2
ifu = 0.
D . Gabai's Results
Theorem 1.2.1 and Theorem 1.2.2 below are main results from [22] and [25] proved by D. Gabai.
These theorems are not only applied in this chapter hut also in chapter 2. Recall that a knot in
a solid torus is called an n-bridge braid if the knot can be isotoped in the solid torus to a braid
which lies in the boundary of the solid torus except for n bridges. We first restate D. Gabai's
main result in [22] as follows with more information added in case 2) due to M. Scharlemann
[67].
Theorem 1.2.1 ([22]) Let K be a knot in a solid torus N* with nonzero wrapping number.
Perform m/l-surgery along K in N* and let K' be the core of the sewn solid torus. Then one
of the following must hold:
1) . N*(K,m/l)
is a solid torus. In this case both K and K' are 0 or 1-bridge braids in N*
and N (K, m/l) respectively.
m
2) . N*(K,m/l)
= D x 5 #I(s,r) where L(s,r) is a nontrivial lens space (\s\ > \), K is
2
1
a cabled knot and m/l = rs.
3) . N*(K,m/l)
is irreducible and dN*(K,m/l)
is incompressible.
7
Chapter 1. On Cyclic Surgery
D . Gabai's proof of the above theorem uses sutured 3-manifold theory based on foliations and
introduced in [23] [24]. In [66] M . Scharlemann developed the /3-norm based sutured manifold
theory and reproved several important results on 3-dimensional topology due to D . Gabai. We
give a proof of Theorem 1.2.1 using M . Scharlemann's theory. We refer to [66] for terminology.
Proof. Obviously N* is a if-taut manifold with empty suture on ON*.
Step (i). If N*(K,m/l)
is irreducible and dN*(K,m/l)
is compressible, then it is easy to
see that N*(K, m/l) is a solid torus.
Note that N* can also be obtained by performing surgery on K' with K as the core of the
sewn in sohd torus.
C l a i m 1. i f , i f ' are braids.
Proof of Claim 1. Take a if-taut surface P in N* whose boundary is a meridian of N*
(of course now the geometric intersection of P with K is their algebraic intersection). By [67]
Theorem 9.1, P is taut in the Thurston norm. Hence P is a meridian disk in N*. A boundary
compressing disk in N*(K, m/l) having minimal intersection with i f ' provides a parameterizing
surface in EQ — N* - intN(K) = N*(K,m/l)
- intN(K').
Performing if-taut decomposition
along P with respect to the parameterizing surface, we obtain a if-taut hierarchy of length 1
(N*, i f )
(N* - intN{P), K - intN(P)).
B y [67] Main lemma 9.7, K — intN(P) is a set of boundary parallel and mutually parallel arcs
in N* — intN(P). Hence i f is a braid in N*. Analogously, K' is a braid in
N*(K,m/l).
C l a i m 2. K, K' are 0 or 1-bridge braids.
Proof of Claim 2.
Let P
x
= PC]E .
0
Then (Pi,dPi)
C (E ,dE )
0
0
is a planar surface
whose u> (winding number of K) inner boundaries (fat vertices), P\ f)dN(K),
orientation induced from Pi and K.
all have the same
Note that the inner boundaries of P\ are meridians of
8
Chapter 1. On Cyclic Surgery
K and P\ has only one boundary component on dN*. Analogously, there is a planar surface (QudQi) C (N*(K,m/l) - intN(K')),d(N*(K,m/l)
- intN(K'))) with exact one outer
boundary component on d(N*(K,m/l) and with all inner boundaries (fat vertices) having the
same orientation induced from Q\ and K'. Note that Eo = N*(K,m/l) — intN(K') and thus
Qi can be viewed as a proper surface embedded in Eo with all inner boundaries having the
surgery slope.
Now the proof of Claim 2 proceeds exactly as in [22] Lemma 2.3, using only elementary
combinatorial analysis of the intersection of the two planar surfaces, Pi and Qi. This proves
!)•
Step (ii). If N*(K, m/l) is reducible, then, by [66] Theorem 4.3, K is cabled and the surgery
slope is that of the cabling annulus, i.e. K = C(r, s), a cabled knot of type (r, s)(\r\ > 1, \s\ > 1
and (r,s) = 1) and m/l = rs. By [28] Lemma 7.2, N*(C(r,s),rs)
= D x S^ftL^^).
2
This
proves 2).
Step (iii). If N*(K,m/l) is irreducible and dN*(K,m/l) is incompressible, we have 3).D
To prove Theorem 1.1.4 we need another result of D. Gabai concerning surgery on knots in
a solid torus, namely
Theorem 1.2.2 ([25] Lemma 3.2)
Let K be a knot in a solid torus N*. If K is a 1-bridge
braid, then only the surgery with slope ±(t+jw)u>±b
or ±(t+ju>)u;±b±l
on K can possibly yield
a solid torus, where u is the winding number of K in the solid torus, t + ju is the twist numbe
of K with 0 < t < u> — 1 (j being an integer), b is the bridge width of K with 0 < b < u> — 1.
See [25] for the definitions of twist number and bridge width of a 1-bridge braid in a sohd
torus.
Chapter 1.
1.3
On Cyclic Surgery
P r o o f of Theorem
9
1.1.4
By Theorem 1.1.3, we may assume that / = 1
L e m m a 1.3.1
N*(K,m)
is a solid torus.
Proof. We first show that N*(K,
m) is irreducible. Suppose that, on the contrary, N*(K,
is reducible.
Then by Theorem 1.2.1.
and m = rs.
By [28] Corollary 7.3, S (K,rs)
*i(S (K ,
3
2), K is a cabled knot C(r,s) on K* with \s\ > 1
3
* S (K\r/s)#L(s,r).
Hence *i(S*(K,rs))
3
r / s ) ) * 7 r ( I ( 5 , r)). If K* is a torus knot, then Ttx(S\K*, r/s))
m
m)
1
S
± 1 by Theorem 1.1.1;
if K* is a nontorus knot, then by Theorem 1.1.3, wi(S*(K r/s)) ^ 1. Hence ni(S*(K, m)) is a
t
free product of two nontrivial groups, contradicting the assumption that 7 T i ( 5 ( i i ' , m)) is cychc.
3
Hence N*(K,m) is irreducible.
Since iri(S*(K,m)) is cychc, dN*(K,m) is a compressible torus in S (K,m).
S
S (K,m)
be a compressing 2-cell for dN*(K,m).
3
Since K* is nontrivial, B
2
C
Let B
2
C
N*(K,m).
Performing 2-surgery on dN*(K, m) using 5 , we get a 2-sphere which must bound a 3-ball in
2
N*(K,
m). Hence N*(K,
m) is a solid torus.
•
By Lemma 1.3.1 and Theorem 1.2.1. 1), i f is a 0 or 1-bridge braid in N*. Note that u> ^ 0
and u> ^ 1 by the definition of satellite knot.
Let B
2
be a proper meridian 2-cell of N*(K,
HxidN'iK^m))
and [dB ]
2
in . H i ( £ # • ( # , m)). Hence
e keriH^dN'iK,™))
m).
Then [dB ]
2
is a primitive element of
—* Hi(N*(K,m))).
By Lemma 1.2.1 (ii),
Chapter 1. On Cyclic Surgery
10
Since w ^ 0,
S (K,m)=S (K\^)
3
= S i(if*,
3
3
and thus
m
Z^ =
H (S\K m))=H (S\K\
1
>
1
)) = Z | | .
m
Hence (w ,m) = 1.
2
Lemma 1.3.2 if* is a torus knot.
Proof. Suppose that K* is not a torus knot. Then, by Theorem 1.1.3, u> = 1 and thus
2
u — 1, contradicting u> ^ 1. •
Lemma 1.3.3 if is a cabled knot on if*.
Proof. By Lemma 1.3.2, if* = T(p,q), a torus knot. By Theorem 1.1.1, 7Ti(5 (A',m)) =
3
7ri(5 (if*,m/u> )) can possibly be cychc only when m is equal to
3
2
(*)
u pq±l.
2
Suppose that K is not a cabled knot. Then if is a 1-bridge braid in N*. By Theorem 1.2.2,
JV*(if,m) can possibly be a sohd torus only when m is equal to
(**)
±(t +
±6
or
±(t + ju)u±b±
1.
Now it is enough to show that any number from (*) can not be equal to any number from
(**). That is to show that \u pq + 1 ± (t +
2
\u pq + 1 ± (z +
2
JU>)OJ
± b\ > 0, \u pq - 1 ± (t +
± 6 ± 1| > 0 and \u pq - 1 ± (t +
2
inequality. The rest of inequalities can be similarly verified.
2
± b\ > 0
± b ± 1| > 0. We verify the first
Chapter 1. On Cyclic Surgery
11
If \pq±j\ ± O.then \u pq+l±(t+ju)u±b\
= \(pq±j)u ±tu±b+l\
2
> \pq±j\u; -tu-b-l
2
>
2
u - (u - 2)w - (w - 2) - 1 = u> + 1 > 0;
2
If \pq ± j\ = 0, then \u pq + 1 ± (t +
± b\ = | ± tu ± b + 1| > tu> + b - 1 > 0. •
2
Now Theorem 1.1.4 follows from Lemma 1.3.2, Lemma 1.3.3 and Theorem 1.1.2. •
1.4
Proof of Proposition 1.1.1
We may assume that m is positive.
Let JV be a tubular neighborhood of K in M and let N = q~ (N).
3
tntJV
Then E - M(m) -
l
E = M — intN is the m-fold cychc regular unbranched covering associated with the
kernel of the composition iti{E) —• Hi(E) = Z —> Z . Let p,, X C BE = ON be a preferred
m
meridian-longitude pair. Then fi = q (p) is a meridian curve ofdE = dN. 9 (A) C dE = dN
_1
_1
is a set of m disjoint 1-spheres each of which bounds a Seifert surface in E. Let A be one of
these 1-spheres. Then p,, X give a framing pair on dE.
Let K' be the core of the solid torus sewn in when performing the m/l surgery on M
along K and let JV' be a tubular neighborhood of K' in M(K,m/l).
M(K,m/l)
- intN' = E. Since p, C dN' is a generator of H {M(K,m/l))
Claim. (p\U*i(E')) =
and let E' = M
m/l
M{K,m/l).
m
- intN'.
n(E).
Proof of Claim. Let a e *i(E'). Then p»(a) = 0 in Hi(M(K,m/l))
m/l
= Z , p-\N') is a
x
solid torus. Let JV' =
of p : M
Then we may assume
= Z
m
by the definition
Therefore (p|).(>i(.E')) C ker(*i(E) —•* i7 (J5) = Z —• Z ) =
x
7ri(J3). On the other hand both
index m. Hence (p|)»(7Ti(J3')) =
(p|)*(7r (J5'))
7Ti(i5).
1
and
7r (£')
1
ra
are normal subgroups of it\(E) of
•
So we may assume E' = E by basic covering space theory.
Let ^* C dE be a 1-sphere with slope m/l, i.e. [/z*] = m[p] + l[X] in i?i(J5) = Z[p] + Z[A].
12
Chapter 1. On Cyclic Surgery
Then p ( M * ) consists of m disjoint 1-spheres p.*, j = l , . . . , m . \fi ] = [p.] + l[X] in iri(dE) =
_1
m
= (PI).([M] + TO in T r ^ d E ) = H^OE). Hence [A] = hx*]
HiidE) = Z[fi] + Z[\] since
in H\(E). If M ( m ) (M /i) is a homology sphere, then E\{E) = Z and [/2] = [p") is a generator
m
of R~i(E). Consequently M /i (M ) is a homology 3-sphere. •
m
m
Remark. As pointed out earlier, Corollary 1.1.2 gives, in particular, the result that if a knot
in S admits Z -surgery, then the determinant of the knot is 1 or - 1 . The following proposition
3
2
which is a consequence of a result of [10] also provides a necessary condition for a knot in S
3
to admit a i?P -surgery.
3
Proposition 1.4.1
Let K be a knot in S
mial of K, i.e. A (l)
K
and let Ax(t) be the normalized Alexander polyno-
3
= 1,
Aif(r') = A (t).
K
If K admits RP -surgery, then A £ ( l ) = 0
3
where " denotes the second derivative.
Proof. A surgery formula for calculating the generalized Casson invariant, as defined in [10],
of the oriented manifold S (K,m/l)
3
\(S (K,m/l))
3
where s(Z, m) = TJ^iJlm
is given in [10], namely
= ( / / 2 m ) A £ ( l ) - (sgn(m)/2)s(l, m),
- [j/m] - l / 2 ) ( ; 7 / m - [jl/m] - 1/2) is the Dedekind sum of / and
m.
As RP
3
can be obtained by 2-surgery on the trivial knot O , it follows that X(RP )
3
=
± A ( 5 ( 0 , 2 ) ) = - 5 ( 1 , 2 ) = 0. Now suppose that S ( A ' , ± 2 ) = RP . Then 0 = A ( 5 ( A ' , ± 2 ) ) =
3
3
(±1/4)A£(1).
1.5
3
3
•
Examples, Remarks and Open Problems
Example 1.5.1. Fintushel-Stern knots
K.
n
R. Fintushel and R . Stern [21] showed, using the Kirby-Rolfsen calculus, that 9n surgery
on the knot K
n
shown in Figure 1.1 yields the lens space Z,(9n,3n + 1). They also showed
Chapter 1. On Cyclic Surgery
13
\
j i i full twists •
L_
I
Figure 1.1: Fintushel-Stern knots K
n
of S branched over K-^n can be obtained by -1/n surgery on the figure eight knot. By W.
3
Thurston's work [77] the cover is a hyperbolic non-Haken manifold. Hence the knot K is
7n
hyperbolic nonsufficiently large knot by [3] (recall that a knot is sufficiently large if there is an
incompressible nonboundary parallel closed surface in the knot complement, otherwise the knot
is not sufficiently large).
Ki is the (-2,3, 7) pretzel knot (see section 2.4.5 for the definition) which is also hyperbolic
and not sufficiently large (see section 3.3). R. Fintushel and R. Stern have shown (unpublished)
that 19 surgery on K also yields a lens space (see section 3.3 for an amusing verification of
2
this result).
Question 1.5.1. Is there any other K , \n\ > 2, which admits two nontrivial successive integral
n
cyclic surgeries?
Example 1.5.2. Berge-Gabai knots J .°
n
It is a remarkable result shown in [25] that -30 and -31 surgeries on the 1-bridge braid in
a solid torus V with presentation data of winding number 7, bridge width 2 and twist number 4
yield solid tori (D. Gabai mentioned in his paper that J. Berge has also independently obtained
this result).
Chapter 1. On Cyclic Surgery
14
Figure 1.2: Berge-Gabai knots J
n
Embedding V into S as a trivial solid torus, we obtain infinitely many knots J
3
n
(Figure
1.2) in S such that each of J admits two nontrivial successive integral surgeries with slopes
3
n
-30 + 49n and -31 4- 49n. Note that every J
n
is a hyperbolic knot by Theorem 1.1.1 and
Theorem 1.1.4.
Recall that a knot K in S has free period n if there is a periodic transformation T of S
3
3
with order n such that {T} acts on S fixed point freely and leaves Ii setwise invariant. In
3
[35] R. Hartley determined free periods for torus knots and for most of knots of ten or less
crossings. From the proof of Proposition 1.1.1,we see that if aknot K C S admits a nontrivial
3
lens space surgery with slope m/l, |m| > 1, then there is a knot K' C S of free period \m\ such
3
that the knot exterior E of K' is an |m|-fold unbranched cychc cover of the knot exterior E of
K. Further we show
P r o p o s i t i o n 1.5.1 There art infinitely many hyperbolic knots in S
3
of free periods.
15
Chapter 1. On Cyclic Surgery
Proof. .Since there are infinitely many hyperbolic knots admitting lens space surgery (Examples 1.5.1-2), the knot exterior of each of these knots is covered by a knot exterior of a knot
in S with free period. Each of these free periodic knots is hyperbolic. This follows from the
3
following
Lemma 1.5.1 If E
E is a finite sheeted regular covering between two knot exteriors E and
E, of two knots, K' and i f , in S , then K' is a torus knot or a hyperbolic knot or a satellite
3
knot iff if is a torus knot or a hyperbolic knot or a satellite knot respectively.
Proof. First note that the finite covering is actually a cyclic covering [27].
Claim 1. if' is torus knot iff if is.
This is equivalent to say that E is Seifert fibered iff E is. But the later statement is true
by [39] Lemma V I 2.9.
Claim 2. if' is a hyperbolic knot iff if is.
In fact, if i f is a hyperbolic knot, i.e. the interior of E admits a hyperbolic structure, then
the interior of E inherits a hyperbolic structure from E through the finite regular covering and
thus i f ' is a hyperbolic knot. Conversely assume that K' is a hyperbolic knot. If K is not a
hyperbolic knot, then K is either a torus knot or a satellite knot. In the case that K is torus
knot, then by Claim 1, if' is a torus knot, contradicting with the assumption. In the case that
i f is a satellite knot i.e. E contains an essential torus T, then p (T) is a set of essential tori
-1
in E', again a contradiction.
Claim 3. if' is a satellite knot iff K is.
This follows from Claim 1 and Claim 2. •
Finally since the exterior of a nontrivial knot covers only finitely many distinct knot exteriors
by [27] Corollary 1.5, of the above free periodic knots infinitely many are distinct. •
A n immediate consequence is
16
Chapter 1. On Cyclic Surgery
Corollary 1.5.1 There are infinitely many hyperbolic knots in S
3
whose knot groups can
imbedded into knot groups of hyperbolic knots as normal subgroups withfinitecyclic quoti
•
Example 1.5.3. Let T(2,3) be the right hand trefoil knot. By Theorem 1.1.1 the surgery on
T(2,3) with slope 5 yields the lens space L(5,9). By the preceding discussion, there is periodic
knot K' in S whose exterior 5-sheeted covers the exterior of T(2,3) and the 5-sheeted cover
3
of S
branched over T(2,3) is a homology 3-sphere Q. Actually K' is the left hand trefoil
3
knot and Q is the Poincare homology 3-sphere [65]. Similarly the surgery on T(2,3) with slope
7 yields the lens space L(7,9), the periodic knot whose exterior 7-sheeted covers the exterior
of T(2,3) is the right hand trefoil, and the 7-sheeted cover of S branched over T(2,3) is the
3
Seifert homology 3-sphere obtained by -1-surgery on T(2,3) [65].
In fact more can be proved using D . Rolfsen's surgery description of branched coverings,
namely corresponding to each cychc surgery on T(2,3) with slope (6/ ± 1)//, the free periodic
knot whose exterior | 6 / ± 1| sheeted covers the exterior of T(2,3) is the left or right hand trefoil
knot and the |6Z ± 1| sheeted cover of S branched over T(2,3) is the Seifert fiberred manifold
3
obtained by l/l or —1/1 surgery on the right or left hand trefoil knot.
Problem 1.5.1. Find the corresponding periodic knots whose exteriors cover the exteriors of
the knots K
n
and J .
n
The following conjecture was raised in [81].
Cyclic Surgery Conjecture. ([81]) For a nontrivial knot K in S and a nontrivial slope m/l,
3
\ir S (K, m/l)\>4.
t
3
As a consequence of Theorem 1.1.1 and Theorem 1.1.4 we see that the conjecture is true
for torus knots and satellite knots.
Question 1.5.2. The knots IO155 and IO157 are knots with free period 2 (see [35]). Does the
exterior of IO155 (or IO157) 2-sheeted cover a knot exterior?
Chapter 1. On Cyclic Surgery
17
If the answer is yes, then there is a counterexample to the cychc surgery conjecture by [27]
Theorem 1.3.1.
Suppose that a knot i f i n S admits a nontrivial cychc surgery of integral slope m . If i f
3
can be isotoped nontrivially into a sohd torus V i n S (i.e. K is not isotopic to the core of V
3
and i f is not contained in a 3-ball of V) such that m-surgery on V along i f yields a solid torus
again, then by Theorem 1.2.1, i f is a 0- or 1-bridge braid in V. If i f is a 0-bridge braid, then
i f is a torus knot or cabled knot i n S . If i f is a 1-bridge braid, then by Theorem 1.2.2 and by
3
presentation of 1-bridge braid i n a solid torus, it can be shown that |m| > 4.
Q u e s t i o n 1.5.3. Let i f C S be a hyperbolic knot which admits a nontrivial cychc surgery
3
with slope m . Can i f be isotoped nontrivially into a solid torus V i n 5 such that m-surgery
3
on V along i f yields a solid torus again? (all known knots in S that admit cychc surgery have
3
this property.)
If the answer is yes, then the cychc surgery conjecture has a positive answer.
Chapter 2
On Property I
2.1
Introduction
Problem 3.6 (D) in [44] asks whether there is a homology 3-sphere which can be obtained by
surgery on an infinite number of distinct knots in S .
3
Examples of homology 3-spheres which
can be obtained by surgery on two or finitely many distinct knots in S
[65] [13] [52] [53].
3
have been given [47]
In a remark to Problem 3.6 (D), R . C . Kirby points out that the Poincare
homology 3-sphere seems only obtainable from -|-l-surgery on the right hand trefoil knot (or,
reversing orientation, from —1-surgery on the left hand trefoil knot). This chapter is devoted
to provide evidence to support this observation.
Note that the fundamental group of the Poincare homology sphere is the binary icosahedral
group, denoted by Ji2o- It has order 120 and its abehanization is trivial. So far it is not known
if the Poincare sphere is the only closed 3-manifold with fundamental group i i o - We call a
2
closed 3-manifold M a fake Poincare sphere if ix\(M) = 7i2o and M is not isomorphic to the
Poincare sphere.
Definition. A knot K in S has property I if every surgery along K does not yield a manifold
3
M with
7Ti(M)
=
ii2o- A knot K in S has property I if every surgery along i f does not yield
3
the Poincare sphere.
Of course the trefoil knot does not satisfy property I.
Conjecture I (I). Every nontrefoil knot in S has property I (I).
3
18
Chapter 2. On Property I
19
Recall that the property P (P) conjecture states that every nontrivial surgery along a nontrivial knot in 5
3
does not yield a homotopy 3-sphere (the 3-sphere). The property P conjecture
was proved recently in [30]. It is known that if the fundamental group of a homology 3-sphere
is finite then it is either the trivial group or else the group J120 [43]. Therefore property I and
property P together are equivalent to the property PI defined as follows.
Definition.
A knot K in 5
3
has property PI if every homology 3-sphere obtained by a
nontrivial surgery along K has infinite fundamental group.
C o n j e c t u r e PI. Every nontrivial nontrefoil knot in 5
3
has property PI.
We wiU prove that the following classes of knots have property I: nontrefoil torus knots,
satellite knots, nontrefoil generalized double knots, periodic knots with some possible specific
exceptions, amphicheiral strongly invertible knots, families of pretzel knots; and that the following classes of knots have property I: slice knots and a certain families of knots formed by
band-connect sums.
Much research has been carried out to prove property P (a list of papers is given in [44]
for research done before 1978, papers thereafter are [61] [50] [51] [74] [63] [20] [8] [9] [62] [1]
[16] [75] [76] [14] [22] [30]). No literature, however, has been found dealing specifically with the
generalized problem we just raised above. As we will see, property P and property I (I) have
certain connections and common features; some techniques which work for property P can also
be generalized to work for the case of property I (I). However in general the two properties do
not imply each other. Certain knots (e.g. slice knots) are found to have property I but are not
known whether or not to have property P. In many cases property I seems a harder problem.
We mainly deal with property I (I) but also include property P when brief arguments apply.
The rest of this chapter is organized as follows. In the next section we briefly introduce some
3-manifold invariants and link invariants namely the Casson invariant, the Rohlin invariant, the
A r f invaxiant, the Conway polynomial and the Kauffman bracket polynomial. These invariants
have apphcations to the property PI problem. In section 2.3 we give, besides a list of known
Chapter 2.
20
On Property I
facts about the Poincare sphere, a complete classification of cyclic group actions on the Poincare
sphere with 1-dimensional fixed point sets. In section 2.4, we prove property I or I for the classes
of knots listed above. The last section consists of remarks and open problems.
2.2
Preliminaries
2.2.1
The Casson Invariant and Property I (I)
In 1985, A . Casson introduced an integral invariant for oriented homology 3-spheres. We briefly
review the representation space construction of the Casson invariant for an oriented homology
3-sphere. For details we refer to [1].
Let M be an oriented homology 3-sphere and let M = V\ U
V i fl V% = dV\ = dV = F
2
be a Heegard splitting, where V i and V2 are handlebodies of the same genus g and F is their
common boundary surface. Let F* be F punctured once. Then the diagram of inclusions:
Vi
\
/
F*
—>• F
M
\
/
V
2
induces the following diagram of surjections on their fundamental groups:
/
H\F*
•
\
7r F
TTiAf.
X
/
\
*iV
2
21
Chapter 2. On Property I
For any group G, call R(G) = Hom(G,SU(2,C))
the representation space from G to
SU(2, C), the 2 x 2 special unitary group. Then the above diagram i n turn induces the following
diagram of injections on representation spaces:
s
\
\
/
R(*iV )
2
Let R(TTiF) d C R(ir\F) be the set of reducible representations, i.e. the set of homomorre
phisms from iriF to SU(2,C) with abelian images. Similarly define R{TC\Vi) d and R(-K\M) dTe
re
Let
Q = image of RfaM)
i n R* =
RfaF*),
Qi — image of R(iriVi) i n R*,
R = image of R(it\F) i n R",
A =image of R(ir\M) d i n R*,
Te
Ai = image of R(^iVi) d i n R*,
re
B = image of R(it\F) d in i i * .
Te
Then i i — 5, Qi - Ai are open manifolds on which SU(2, C)/center acts freely by conjugation.
Let
Q = Q — A modulo action by conjugation,
Qi = Qi — Ai modulo action by conjugation,
R = R — B modulo action by conjugation.
Then Qi, i = 1,2, embed properly in R and their intersection is compact. Furthermore Q
2
can be moved by an isotopy in R to Q such that Q i and Q intersect transversally at finitely
2
2
Chapter 2.
22
On Property I
many points i n R. The orientation of M can be used to determine an orientation of Qi, R", Qi
and R. Therefore an algebraic intersection number < Q\,Q
Also note that R* is a manifold isomorphic to ( S ) *
3
5
2
>fi
Q11Q2 >ft
=<
c a n
and Qi C R*, i = 1,2,
submanifolds of middle dimension, both being isomorphic to (S ) . Let < Qi,Q
3
3
denned.
D e
are compact
> « • be the
2
homological intersection number of Q\ and Q i n R*. Then Casson invariant of M, denoted by
2
A, is given by
A . Casson proves that this number is an integer and is independent of the Heegard decomposition of M. Note that | < Q\,Q
2
>R* \ - \R\(M)\ = 1, therefore < Qi,Q
2
>R is an even
integer. A n immediate consequence of the construction is
Theorem 2.2.1
( A . Casson) (i). A ( - M ) = - A ( M ) , where -M
denotes opposite orientation
ofM.
(ii). \{M) = 0 i/iri(Af) = 1.
The Casson invariant can also be computed very effectively by a surgery formula.
Theorem 2.2.2
M(K,\[l)
( A . Casson) Let K be a knot in an oriented homlogy 3-sphere M and let
be the homology 3-sphere obtained from M by performing 1/l-surgery on K.
AA-(t) be the normalized Alexander polynomial of K, i.e. A j f ( l ) = 1 and A j c ( t ) = A t f ( f ) .
-1
Then
A(M(/f,l/0)=A(M) + /(l/2)A^(l).
where A ^ - ( l ) ts the second derivative of Ax(t)
valued at 1.
For a knot K in S we shall call X'(K) = (1/2)A'£(1) the Casson invariant of K.
3
Let
23
Chapter 2. On Property I
Let T denote the right hand trefoil knot and D the Poincare homology sphere. Since D can
3
3
be obtained by 1-surgery along T and A (t) = - 1 +1 + r " , we have A(I> ) = ( 1 / 2 ) A £ ( 1 ) = 1.
1
T
Now suppose that S (K,l/l)
is the Poincare 3-sphere obtained by 1/i-surgery along a knot
3
K in S .
3
3
Then by Theorem 2.2.2, X(S {K,l/l))
= / ( 1 / 2 ) A £ ( 1 ) = 1 or - 1 . It is known
3
that the normahzed Alexander polynomial of any knot K in S can be expressed as A.R-(*) =
3
Ei=i
a +
0
+ *"*') € Z[t,t~ ). B y a simple calculation we get ( 1 / 2 ) A £ ( 1 ) = £ J = J i
l
a
{2
€ Z.
Therefore / = 1 or - 1 and ( 1 / 2 ) A £ ( 1 ) = 1 or - 1 . This simple observation gives
Lemma 2.2.1 LetK be a knot in S . IfS (K, l/l) is the Poincare sphere, then X(S (K/l/l)
3
3
=
3
1 or - 1 , / = 1 or - 1 and X'(K) = (1/2)A^(1) = 1 or - 1 .
Proposition 2.2.1 There are exactly two irreducible representations from 7j2o to SU(2,C) up
to conjugation in 517(2, C).
Proof. 7i2o has the group presentation {x,y;x = (xy) = y ,x
2
3
s
Let p : 7i2o —> SU(2,C) be an irreducible representation.
4
= 1}.
Note then p(/i o) must be a
2
non-abelian subgroup of 517(2, C).
(
1). Claim. p{x) =
2
Proof of Claim. (p(x) )
2
2
= p(x) = p(x ) =
A
4
, 1 0 ,
|
v0 1
|. Thus the eigenvalues of p(x)
2
1
are either 1 or - 1 . Consequently p(x) — j
2
1
1
0
] or p(x) = j
1 /
V 0
/
P(y)
5
=
2
/
0 .
, then p(x)
0
0 \
| or p(x) =
0 1
and thus p(y) =
o
1
p(Zi2o) is non-abelian.
1
0 .
0
1
|. Therefore p(xy) = ±p(y)
3
3
=
-1
But this contradicts the assumption that
Chapter 2. On Property I
24
0 1
2). Note that
-1 0
p(y) = |
h
0
e&
0
A"
0 -1
1
1
L - l
0
0
/
-1
I
0
or |
A
0
|, p(y) is conjugates to
0 -»
/
,
j and p(xy) is conjugates to
e~*f~
0
Since p(x) = p(xy)3 _
2
I i
I, it follows that p(x) is conjugate to |
0
0
0
0 \
, -1
e?
A
0
e-""'/
3
In particular the trace of p(xy) is tr(p(x)p(y)) = e " / + e " * / = 1.
3
-
e
3). After a conjugation, we may assume that p(y) =
(x) e {B
\
I
IB^-B 6 SU{2,C)} = {[
°
0 -i
P
j
V
**
^
-1
2 * t n5 2 I
and |6|2 = 1 — t = 1 — . •
2
solution set b =
C2e «,
e
\„,.
s
0
0
e~~
, n = 1 or 3. Then
| ; i e 7 2 , 5 e C , t + |&| = l}.
2
\ - 6 -rz
From 2) we conclude that 1 = tr(p(x)p(y)) = tie^
3
2
— tie "s^" = —2tsin~- and thus t =
A s c > 0, |6|2 = c has
Let c — c(n) = 1 — .
0 G [0,27r).
Hence we may further assume that
/
{p(x),p(y)} = {
h~6i
C2e
-ck e
PI
cT
_
£
~2^f
0
0
s
e~~
0
0i
2sm
e
0 2
2st'n^
e
nm
e~T
s
0
e
s
0
0
e£
0 may
e~finally assume that
Thus we
and p(y) =
_ C 2
217^ /
e ~
n
0
-0X1.
, n = 1 or 3.
0 e s
Consequently, there are, up to conjugation, at most two irreducible representations p :
7
120
—*SU(2,C).
25
Chapter 2. On Property I
for
It is easy to check that the preceding p(x) and p(y) satisfy (p(x)p(y)) =
3
0
-1
n = 1 and 3, and they define two representations p '• /120 — • SU(2,C), n = 1,3. pi and />3
n
are not equivalent since |
ei
0
I
/ e s
| and |
0
, 1
have different traces. •
If M is a closed 3-manifold with fundamental group I\2o, then Q — Q\ n Q in the con2
struction of the Casson invariant given at the beginning of this section consists exactly of two
points by the above proposition. We do not know if Q\ and Q intersect transversally at these
2
two points. But after an isotopy we can only have < Q i , Q2 >= 0
o
r
i l - Hence we have, using
Casson's surgery formula again,
Lemma 2.2.2 If^S ^,
3
1//) = I , then X(S (K, 1//)) = 0 or ± 1 . Therefore X'(K) = 0 or
120
3
±1.
2.2.2
T h e Rohlin Invariant and the A r f Invariant
Here we consider the relation of the Casson invariant with the Rohlin invariant and the A r f
invariant and its consequences for property I and property P.
Recall that the Rohlin invariant p of a homology 3-sphere M is defined by p(M) = o~(W)/8
mod 2, where W is a simply connected 4-manifold with even quadratic form and with M as
boundary, and cr(W) is the signature (index) of W . Also recall that the Z2-valued Arf invariant
4
a of a knot K in S is defined by a(K) = ]£»=i V2t-i,2i-i«2i,2t
3
m o c
* ^'
w
n
e
r
e
is
a
2n X 2n
Seifert matrix for K [46] [58] [64] [40]. In [26] F . Gonzalez-Acuna established a surgery formula
for calculating the Rohlin invariant of homology 3-sphere obtained by 1//-surgery on any knot
K in S , that is
3
Theorem 2.2.3 ([26]) p(S (K, 1//)) = la(K) mod 2.
3
The Casson invariant and the Rohlin invariant are related as follows.
26
Chapter 2. On Property 1
Theorem .2.2.4 ( A . Casson) Let M be a homology 3-sphere. Then p(M) = A ( M ) mod 2.
Corollary 2.2.1 Any knot K in S
3
of Arf invariant 0 has property I.
Proof. By Theorem 2.2.4 and Theorem 2.2.3, \{S (K, If I)) s p(S (K, 1//)) = la{K) = 0
3
3
mod 2. Hence S (K, 1//) cannot be the Poincare homology sphere by Lemma 2.1. •
3
Similarly the following corollary follows from Theorem 2.2.4, Theorem 2.2.3 and Theorem
2.2.1 (ii).
Corollary 2.2.2 Any knot K in S
3
of Arf invariant 1 has property P.
Note that for a knot K e 5 , A'(Ji') = a(K) mod 2 by Theorems 2.2.2-4.
3
2.2.3
T h e Conway Polynomial and the Kauffman Bracket Polynomial
These two polynomial invariants of links shall be used in section 2.4.2 and section 2.4.4. Here
we only give their definitions and some properties which we will use.
The Conway polynomial invariant [15] is defined by the following three axioms.
Axiom 1). To each oriented link L in S there is an associated polynomial ^L(Z) G Z[Z].
3
Ambient isotopic links have identical polynomial.
Axiom 2). Vv = 1 where U denotes the unknot.
Axiom 3). V s ^ ( r ) - V-^(z)- zV^z) = 0, where >£, X and X s t a n d for oriented links which
look like that in a neighborhood of a point and identical elsewhere.
Remarks:
(i) . I f L is a knot, then
is independent of the choices of orientations for L.
(ii) . Let L* denote the mirror image of L. Then Vjf(z) =
(iii) . Let V i ( z ) = ao + a z +
x
h az
n
n
Vi(-z).
be the Conway polynomial of a link L. Then
27
Chapter 2. On Property I
lk(L)
di = 0
if L has two components,
otherwise.
where lk(L) denotes the linking number of L.
(iv). If £ is a knot, then VL(< ' -I" ' ) = AL(I), where Ar,(t) is the normalized Alexander
1/ 2
1/ 2
polynomial of I, i.e. AL(1) = 1 and A L ( I ) = AL(I)-1
In [41] L. Kauffman reconstructed the Jones polynomial through his bracket polynomial. The
Kauffman bracket polynomial < L > (A) G Z [ A , A ] is denned for unoriented link diagrams
_ 1
L, with the following defining relations.
1) . < X >=
A
<~>
<)(>» <><>= A " <x> +A <)(>,
1
where >< , X , ~ , )( stand for links which look like that in a neighborhood of a point and
identical elsewhere.
2) . < O >= 1, < O U L >= ( - A - A" ) < L >,
2
2
where O is the unknot diagram with no crossing points and U is the disjoint union.
< L > (A) is not a link invariant but it can be adjusted to be one for oriented links under
ambient isotopy. Given an oriented link diagram L. Let w(L) be the algebraic sum of the
crossings of L, counting X, and X as +1 and —1 respectively. Then
f (A) = (-A)- ^
3
L
<L>(A)
is a desired invariant of oriented links under ambient isotopy. We shall call /L(A) the oriented
Kauffman bracket polynomial.
.Remark:
(a) . If L is a knot then /L(A) is independent of choices of orientations.
(b) . Let X* denote the mirror image of L. Then fa* (A) = / L ( A ) .
- 1
(c) . /L(* ^ ) is the Jones polynomial.
-1
4
28
Chapter 2. On Property I
We leave the well definedness of the Conway polynomial and the Kauffman bracket polynomial and proofs of the remarks to the reference [15] [41] [42].
2.3
Cyclic Actions on the Poincare Homology 3-Sphere
In this section we give a complete description of orientation preserving isometric cychc actions
on the Poincare sphere D . Combining with a result of Thurston's, we give a classification of
3
cychc actions on D withfixedpoint sets of dimension 1.
3
The Poincare sphere,firstconstructed by Poincare, is a very special manifold. It seems to be
the first known example of a nonsimply connected closed 3-manifold with trivialfirsthomology
group. So far it is the only known homology 3-sphere with nontrivialfinitefundamental group.
Let D denote the Poincare sphere. Several descriptions of D can be given as follows,
3
3
1) . the manifold with the surgery presentations shown in Figure 2.3;
2) . the Seifert manifold with 3 exceptionalfibersof type (5,1) (3,1) and (2,1), and crosssection obstruction —1;
3) . the Brieskorn manifold {(z z , z ) G C ; z\ + z\ + z\ = 0, |*i| + |z | + N
u
2
3
3
2
2
2
2
= 1};
4) . the quotient space of S under the free action of the binary icosahedral group, J120 =
3
{x,y;x = (xyj = y , x = 1}. Hence the universal cover of D is S and the fundamental
2
3
5
4
3
3
group of D is J120;
3
5) . the space constructed from a regular dodecahedron by identifying each boundary point
with the point on the opposite face rotated 36° about the axis perpendicular to the faces, in a
clockwise sense;
6) . the 2-fold (3-fold, 5-fold) cychc branched cover of S branched over the (2,3) ((2,5),
3
(3,5)) torus knot.
7) . the boundary of the 4-manifold obtained by plumbing on the Es weighted tree.
30
Chapter 2, On Property I
For more details see [65] [45].
The following lemma will be applied.
Lemma 2.3.1 Let X be a path connected, locally path connected and semilocally simply co
nected space and let p : X —• X be a universal covering projection. Let G be a group of
isomorphisms of X and let T be the group of covering transformations of p. Define G = {g
g : X —• X a map with pg = gp for some g £ G}. Then
1) . G is a group of isomorphisms of X;
2) . if N C G is a normal subgroup, then N C G is a normal subgroup. In particular,
T = {1} C G is a normal subgroup;
3) . for each g £ G the element g £ G with pg = gp is unique, the map p. : G —-»• G define
by p*(g) ~ g is an epimorphism, and the sequence
r —-» G
1 —>
G —• 1
is exact.
Proof. First note that for each element g £ G there exists an element g £ G such that
pg = gp. In fact, let g £ G, x £ X, x £ p (x) and y £ p~ (g(x)). By basic covering space
-1
1
theory, there is a map g : (X,x) —• (X,y) such that the following diagram commutes.
(X,x)
-L>
(X,y)
Pi
Pi
(X,x)
(X,g(x))
i.e. pg - gp.
1). Let § 52 6 G. Then pg^ = g pg = gigiP, i.e. g\h 6 G.
u
x
2
Chapter 2.
31
On Property I
Let g G G, i.e. pg = gp for some g G G. Let x G X, x = p(x), y = g(x) and y = g{x) = p(y).
Then by the note above there exists an element g' G G with pg' = g~ p
l
But pgg' = gg p
-1
and with g'(y) = x.
= p and gg'(y) = y. Hence gg' = 1. Similarly, g'g = 1. Therefore 5 ' =
We hence proved that each element of G is an isomorphism of X and G is a group.
2 ) . Let h G N and let 5 G G . Then pghg' = ghg~ p, i.e. p ^
1
l
-
1
G iV.
3) . For the uniqueness, note that if pg = gp and pg = fp for g,f
£ G> then 47) = / p and
then <7 = / since p is an onto map.
B y the preceding remark, p» is an onto map. It is easy to check that p , is a homomorphism.
Obviously fcer(p«) = T. Therefore the sequence
1 —• T — * G
G —• 1
is exact. •
In the following theorem we present D
3
orthogonal action of 50(4) on 5
3
as 3-dimensional space form, i.e. consider the
and let D = 5 / / o where /120 is a subgroup of 5 0 ( 4 ) . If
3
3
12
(-Ti2o)i> (^120)2 C 5 0 ( 4 ) are subgroups isomorphic to 7i o, it follows from [68] Theorem 4.10
2
and Theorem 4.11 that they are conjugate in 0(4).
Consequently 5 /(/i2o)i and 5 /(/i2o)2
3
3
are isometric. Thus D = 5 /7i2o is independent of the choice of the subgroup /120 C 5 0 ( 4 ) .
3
T h e o r e m 2.3.1
3
(i). For each integer n > 1 there is an orientation preserving isometric Z
n
action on the Poincare 3-sphere 5 //i203
(ii) . Up to conjugation by an isometry, such a Z
n
action is unique for each n.
(iii) . If n is relative prime to 2,3 and 5, then the Z
n
2,3, or 5, then exactly those elements of Z
n
action is free; if n is not prime to
which have orders 2, 3 or 5 have fixed point se
of dimension 1.
Proof. The basic reference for the facts stated in the proof is [68].
Chapter 2. On Property I
32
Consider the following exact sequence ([68] p.453)
1 —• Z —» 5 0 ( 4 ) -1+ 5 0 ( 3 ) x 5 0 ( 3 ) —• 1.
2
Let ieo be a subgroup of 5 0 ( 3 ) isomorphic to the icosahedral group. Let 7i2o = »? (^60 X !)•
-1
Then ii o C 5 0 ( 4 ) is isomorphic to the binary icosahedral group and acts on 5 fixed point
3
2
freely by isometries. We shall take p : 5 —• D = 5 /7i o as a standard universal covering of
3
3
3
2
the Poincare sphere D .
3
(i) . We first prove the existence. Let {/")
/ € n
- 1
C 5 0 ( 3 ) be a cychc group of order n. Let
( l X /") and let F be the subgroup of 5 0 ( 4 ) generated by 7i o and / . Note that /
2
has order n , / 7 i o /
- 1
2
= A20 and F is a group of isometries having 7i o as a normal subgroup
2
of index n . Let 7i o act on 5
2
3
first and thus get the quotient space D . There is an induced
3
orientation preserving isometric cychc action on D of order n as follows: Let p : S —• D
3
3
be the covering projection corresponding to the 7i o action and define / : D
2
f(x) = pf(x) where i G D
3
3
—• D by
3
and x £ p ( x ) . Then / is well defined; i n fact, let x' £ p~ (x),
- 1
3
l
then there is a 6 7 o such that ot(x) = x ' and thus pf(x') = pf(a(x)) — p@f(x) = pf(x) where
l2
(3 = faf-
1
£ I .
120
Similarly, using / - * , define / ' : D —* D by f'(x) = p / ( * ) - I t
3
3
_ 1
i s
easy
to check / ' / = 1 and / / ' = 1, and thus / is an isometry of D . As fp = pf, the order of / is n .
3
(ii) . We now pTove the uniqueness (up to conjugation by an isometry). Let g : D —• D
3
3
be an orientation preserving isometry of order n . We may assume that the geometric structure
on D is induced from the universal covering p : 5 —• D given at the beginning of the proof.
3
3
3
We shall prove that, up to a conjugation by an isometry of D , the {g} action is equivalent to
3
the {/} action given in (i).
Let G ={g;g : 5
3
—• 5
3
a map with pg = g p for some integer k}. Then G C 5 0 ( 4 ) by
k
our construction. B y Lemma 2.3.1, 7i o C G is a normal subgroup of index n . More explicitly,
2
G = Ufc~i 9 A20 for some g £G with pg = gp.
Claim 1 . There is an element h £ 5 0 ( 4 ) such that hGh'
1
= F.
Chapter 2.
33
On Property I
Proof Of Claim 1. Still consider the exact sequence
1
Z
— >
2
— •
50(4)
50(3) x 50(3) —+ 1.
Let p,, i = 1,2, be the natural projections from 50(3) X 50(3) to its left and right 50(3) factors
respectively. Then we must have p\n(G) = Ieo since 50(3) has no finite group containing /so
as a proper subgroup. Let n(g) = g' x g" G 50(3) X 5 0 ( 3 ) . Then g" £ 1 since otherwise the
kernel of n would be larger than Z . As g' G Ieo,
2
(ff'
_ 1
x \){g' x g") = 1 x g" G r/(G). Suppose
5" has order m in 5 0 ( 3 ) . Then we see n(G) = ho X {g") and thus m = n by Lemma 2.3.1.
Since isomorphic subgroups of 5 0 ( 3 ) are conjugate, there are h" G 5 0 ( 3 ) such that
h"{g"Yh"- = {/"}. So n(G) is conjugate to Ieo x {/"} i n 50(3) x 5 0 ( 3 ) by the element
1
1 X h". Let h G r / ( l x h"). Then since the kernel of n is Z which is contained in both G and
- 1
2
F, hGh.- = F. Note that hl^oh'
1
= Iuo-
1
C l a i m 2. There is an isometry h : D —• D such that /i{<7}/i = {/}.
3
_1
3
Proof of Claim 2. Define h : D —• D by /i(x) = ph{x) where x G p~ (x). Then / i is
3
3
l
weU defined. In fact, let x . G p ( x ) , then there is a G /120 such that a ( x ) = x» and thus
_1
ph(x»)
— ph(a(x)) = p/3h(x) = ph(x) where (3 = / m / i
h' : D — * D by /i'(x) = p h
3
3
- 1
^ ) where x G p
- 1
^)-
- 1
G 7i o- Similarly, using h' , define
1
2
I<; i s e a s
y
t o c h e c k
=
1
a
n
d h h >
=
and thus / i is an isometry and h~ = h'.
x
Now let x G £ , hgh-*(x) = hgph,- ^) = hpghr (x) = p / ^ - ^ x ) = p / . ( x ) = / * p ( * ) =
3
1
l
/ ( x ) where / , = hgh' G F has order n . Hence /
fc
1
f c
has order n and thus
= {/}.
(iii). Note that 50(3) x 50(3) is the orientation preserving isometry group of 50(3) and
the following diagram
50(4) x 5
50(3) x 50(3) x 50(3)
3
1
I
S
3
9
50(3)
34
Chapter 2. On Property I
commutes, where the two vertical arrows denote the actions on 5 and 50(3) respectively and
3
q is the quotient map defined by the standard Z action on S .,
3
2
Note that an element g' x g" 6 50(3) X 50(3) acts on 50(3)fixedpoint freely iff g' is not
conjugate to g" in 50(3). Also note that two elements in 50(3) withfiniteorders can possibly
be conjugate only when they have the same order. Hence if n is relative prime to 2,3,5, then
any element in J o x {g"} acts on 50(3) freely since orders of elements in 7 o can only be 2,3
6
6
and 5. Hence we have a free induced Z action on D .
n
3
If n is not relative prime to 2,3 or 5, then exactly those elements c' X c" 6 I&o x {g"} with
c' and c" having orders 2,3, or 5 and being conjugate to each other have fixed point sets in
50(3). Such elements exist. Hence in these cases, we obtain ^60 X Zk,k = 2,3 or 5, actions
on 50(3) withfixedpoint sets. This in turn induces orientation preserving Zk actions on D
3
with fixed point sets. By Smith theory [11] thefixedpoint set of each such cychc action is a
1-sphere in D . •
3
Corollary 2.3.1 Let g : D —• D be an isomorphism of order n. If thefixedpoint set of
3
3
g has dimension 1, then n = 2,3 or 5 and such action is unique up to a conjugation by an
isomorphism of D .
3
The proof of Corollary 2.3.1 is based on the following W. Thurston's result which will also
be applied later on.
Theorem 2.3.2 ( W . Thurston) Let M be an irreducible closed 3-manifold which admits a
finite group action withfixedpoint set of dimension 1. Then M has a geometric decomposition.
Furthermore if M is also atoroidal, then M admits a geometric structure such that the grou
action is by isometries.
Proof of Corollary 2.3.1. By Theorem 2.3.2, we may assume that / is an isometry. Note
that / is necessarily orientation preserving since it has 1-dimensionalfixedpoint set. Now apply
Theorem 2.3.1. •
35
Chapter 2. On Property I
Results in this section will be applied in sections 2.4.3-4.
2.4
Knots Having Property I or i
2.4.1
Torus knots, Slice Knots and Knots Formed by Band Connect Sums
In this section we show property I for nontrefoil torus knots and property I for slice knots and
a family of knots formed by band-connect sums.
Proposition 2.4.1 Nontrefoil torus knots have property I.
Proof. This proposition is implicitly contained in [56]. Here we give a proof using the
Casson invariant. Let T(p, q) denote the torus knot which wrapps around the boundary of an
unknotted solid torus p times meridianly and q times longitudely. Note that (p, q) = 1 and
we may assume that p > q > 0. If q = 1, then T(p, 1) is the trivial knot which obviously has
property I. So we may assume p > q > 1. Note also that T(3,2) is the trefoil knot and hence
to be nontrefoil, p ^ 3 or q ^ 2.
It is known that the Alexander polynomial of T(p,q) is A(t) =
Ait- ) =
1
I-(P- )(9- )A(I),
1
1
G Z[t]. Since
we have to normalize A(r) to A{t) = f ~ ( p "V ( ''" 1 ) A(t). Pure calcula-
tion of the second derivative of A(t) gives (1/2)A"(1) = (p - l)(c - l)/24. Since p > q > 1
2
2
and p ^ 3 or q ^ 2, (1/2)A"(1) > (3 - 1)(2 - l)/24 = 1. Now apply Lemma 2.2.2 •
2
2
Corollary 2.4.1 ([37]) Nontrivial torus knots T(p,q) have property P.
Proof. As ( l / 2 ) A £
(pij)
( l ) = (p - l)(g - l)/24 ^ 0 for p > q > 1, Theorem 2.2.1 (ii)
2
2
applies. •
Proposition 2.4.1 and Corollary 2.4.1 together give
Corollary 2.4.2 Nontrivial nontrefoil torus knots have property PI. •
36
Chapter 2. On Property I
Figure 2.4: a band-connect sum of two knots
Proposition 2.4.2
Slice knots (and hence ribbon knots) have property 7.
Proof. It is known that A r f invariant is an invariant of concordance [64]. Since any slice knot
is concordant with the trivial knot and the A r f invariant of the trivial knot is 0, Proposition
2.4.2 follows from Corollary
2.2.1. •
Now we show that if two knots have the same (different) A r f invariant (invariants), then the
knot formed by band-connect sum of the two knots has property I (P). The argument is based
on Kauffman's geometric version of A r f invariant as well as results in section 2.2.
Let Ki and K
2
be knots in S .
follows. Separate Ki and K
imbedding such that
The band-connect sum of K\ and K
3
2
2
by an imbedded 2-sphere S
2
6 (7i' ) = 7 x 0 ,
_1
1
6 (ijf ) = 7 x 1 .
_1
C S.
3
is a knot denned as
Let 6 : 7 x 7 — • S
Then join the arcs K
x
2
3
be an
- 6(7 x 0) to
K2 — 6(7 x 7) by the arcs 6(97 x 7). The resulting knot is the band-connected sum of K\ and
K,
2
denoted by Ki#bK
2
imbedded in S
and
3
(Figure 2.4). 6 is called a trivial band if there exists some 2-sphere S
2
such that 6(7 x 7) D S
2
is a single arc, S fl (K\ U K )
2
Note that if 6 is trivial, then Ki# K
Proposition 2.4.3 7/ TiTi and K
K = K\#\,K2 has property I.
b
2
2
=
2
— 9 and S
2
separates Ki
K #K .
X
are two knots in S
2
3
having the same Arf invariant, then
37
Chapter 2. On Property I
Note that if b is trivial and K\, K are both nontrivial then K = Ki#K
2
and thus has property I by Proposition
2
2.4.6.
is a satellite knot
Note also that if b is trivial and one of two knots,
say Ki, is trivial, then K = K has property I by Corollary
2
2.2.1.
To prove Proposition 2.4.3 several lemmas are needed.
Lemma 2.4.1 Let K = Ki#K
be a composite knot in S . Then A*:(0 = A*r (t) • A (t)3
2
2
K2
Proof. Let F, be a Seifert surface of Ki with genus g,, i = 1,2. Then F = Fi\F (the
2
boundary connect sum) is a Seifert surface of genus g + g of K. The normalized Alexander
x
polynomial of K is A K ( « ) = t-^ Uet(V
- tV ) where V is a Seifert matrix of F. Obviously
1+3i
V =
V i
° j where V,- is a Seifert
T
matrix of Ki for i =
tV ) = t~^det(Vi - tVi ) • t~ *det{V - tV ) = A (t)
T
r
3
2
T
2
2
Kl
1,2.
Hence A (t) =
K
t~toi +92)det(V-
• Ajr (<). •
2
For any knot K i n S , its normalized Alexander polynomial can be expressed as A;r(t) =
3
ao + £*.•(< - t- )-
Thus A' (t) = £ a , ( l - i ~ ) and A f c ( l ) = 0. If K = Ki#K ,
1
by Lemma
2.4.1,
2
K
then
2
Afc(<) = A ^ ( i ) • A * ( * ) + 2A' (t) • A' (t) + A (t)
a
Kl
K2
Kl
• A'^t)
and thus
A £ ( l ) = A ^ ( l ) + A £ ( l ) . Therefore we have
2
Lemma 2.4.2 For a composite knot K = Ki#K
2
By the note given at the end of section
2.2.3,
in S , X'(K) = \'{Ki) + \'{K ). •
3
2
we see
Lemma 2.4.3 If Ki and K are knots in S having the same (different) Arf invariant (invari3
2
ants), then a(Ki#K )
2
is 0 (I). •
Now we apply Kauffman's geometric interpretation of the A r f invariant of a classical knot
[40] to prove
Proposition 2.4.4 Let Ki and K
2
be knots in S . Then a(Ki# K )
3
b
2
=
a(Ki#K ).
2
38
Ch&pter 2. On Property I
r
IJ
Figure 2.5: T-moves
Proof. In [40], L.H. Kauffman denned a Inequivalence relation for knots in S and showed
3
that two knots in S are r-equivalent if and only if they have the same Arf invariant. The
3
T-equivalence is denned as follows. Let A" be a knot in S . Take an oriented knot diagram
3
K (orientation is arbitrarily given). The types of strand-switch of K shown in Figure 2.5 are
called T-moves. Now two knots in S is T-equivalent iff one knot can be deformed to the other
3
by finitely many T-moves as well as knot isotopies. See [40] for more details.
Take a knot diagram oiKi#\>Ki such that K\ and Ki have induced disjoint diagrams which
can be separated by a 1-sphere of the projection plane and such that the band b is thin and
intersects K\ and Ki transversally. Then there are finitely many crossings where the band b
crosses under the knot K\. Performing T-moves on these crossings, we obtain a knot which is
isotopic to the composite knot Ki#K .
2
This process is best illustrated by the example shown
in Figure 2.6. Hence Ki#bKi is T-equivalent to Ki#Ki
and thus they have the same Arf
invariant. O
Proof of Proposition 2.4.3. It follows from Proposition 2.4.4, Lemma 2.4.3 and Corollary
2.2.1. •
Similarly we can prove
Proposition 2.4.5
If K\ andKi are knots in S having different Arf invariants, then Ki#bKi
3
has property P. •
Note that property P for nontrivial band connected sum has been proved by A. Thompson
[75]. Of course property P for an arbitrary knot in S has been proved recently in [30].
3
Chapter 2. On Property I
Figure 2.6: ff,#bffa « r-equivalent to
Chapter 2. On Property I
2.4.2
40
Satellite Knots and Generalized Double Knots
In this section we show property I for satellite knots and generalized double knots.
Proposition 2.4.6 Satellite knots have property I.
Proof. The argument is similar to that of Theorem 1.1.4. We need one more result from
[28], that is
Lemma 2.4.4 ([28]) Let i f = C(p,q) be a cabled knot in a solid torus N*. Then N*(K,m/l)
is a solid torus iff m = Ipq ± 1.
Let i f be a satellite knot in S with if* as a nontrivial companion knot. Let N and
3
N* be tubular neighborhoods of if and if* in S with N C intN*. Let E = S - intN,
3
3
E* = S - intN* and E = N* - intN. Then E = E* U E . Let p, X C dE and p*, X* C dE*
3
0
0
be preferred meridian-longitude pairs of if and if* respectively. Let u> be the winding number
of i f in N*.
Suppose that S (K,l/l)
3
compressible in S (K,l/l)
3
is a manifold with fundamental group I\20- Then dN* must be
by Dehn's lemma. Let (D ,dD )
2
C (S (K,l/l),dN*)
2
3
pressing 2-disc. Since dN* is incompressible in E*, (D ,dD )
2
2
C (N*(K,l/l),dN*).
be a comHence
case 3) of Theorem 1.2.1 is ruled out. Case 2) of Theorem 1.2.1 cannot hold either by our
assumption. Therefore N*(K,lfl)
is a sohd torus and if is a 0 or 1-bridge braid in N*. Hence
w > 1 by the definition of a satellite knot. But by Lemma 2.4.4, i f cannot be a 0-bridge braid
and by Theorem 1.2.2, i f cannot be a 1—bridge braid. A contradiction is thus obtained. •
Proposition 2.4.6 and Corollary 1.1.1 together give
Corollary 2.4.3 Satellite knots have property PI. •
Recall that a generalized double knot is defined as follows. Let V be an unknotted sohd
torus and let if o be the knot contained in V as shown in Figure 2.7 (a). Let if* be any knot
Pi
Chapter
2.
41
On Property J
p full
p>0
^
twists
x
p<0
(b)
(8)
Figure 2.7: a generalized double knot
in S and let A " be a tubular neighborhood of K" in 5 . Let / be an isomorphism from V to
3
N*.
7
3
Then the image K = f(K ,o) of K o under / is called a generalized double knot and K*
p
is called a companion
Py
knot of K = f(K o) (Figure 2.7). Note when p = 1 this is just the usual
p>
definition of a double knot.
P r o p o s i t i o n 2 A.7
Proof.
Nontrefoil generalized
double knots have property I.
Let K be a generalized double knot in S and let K* be its companion knot. If
3
K* is a nontrivial knot, then K is a satellite knot and Proposition 2.4.6 applies. If K' is the
trivial knot, then K is a generalized twisted knot (Figure 2.8). So we assume that A' = A'p.ji
a generalized twisted knot with q twists (Figure 2.8 (a)). Note that K
Pi0
is the trivial knot,
K\-\ is the right hand trefoil knot, A ' _ i , i is the left hand trefoil knot, A " _ i , _ i and A'j,i are the
figure eight knot and /i'o, is the trivial knot.
9
C l a i m . The normahzed Alexander polynomial of K
Proof of the Claim.
Pt9
is
AAy,(0 = 2pg + 1 -
First we calculate the Conway polynomial ^K , (t)
p q
on the number of twists. Orient K
Pi9
of
iv
P l ?
pq(t +
I ).
-1
by induction
arbitrarily. Then by Conway recursion formula, V A ' , , _ I P
Chapter
2. On Property
42
1
P,0
q
full
P,Q-1
p
L
twists
-XT >^CK- XT
XXX
q>0
q<0
( )
(b)
8
<c)
Figure 2.8: generalized twisted knot Ji'
V^-
P i ?
= z V . ^ , where L
p
Pi9
is the link of two components shown in Figure 2.8 (c). The Conway
polynomial of L with the orientation given in Figure 2.8 (c) is pz (again using Conway recursion
p
formula inductively).
^Kp,o
Therefore we get ^K ,
P 9
- qp* = 1 - pqz - Hence
2
2pq -I- 1 - pq(t + i
7
- 1
= VA' ,_
= V^,., - pz
7
|(I
the Alexander polynomial A A ,(r) = 1 p
2
- 2pz = • • • =
2
pq{t ^ - t~^l ) =
1
2
2
2
) and the claim is proved.
Simple calculation gives ( l / 2 ) A £ ^(1) = pq. Hence by Lemma 2.2.2 only when p = ± 1 and
P
q = ± 1 or p = 0 or q = 0 could K
p<q
have chance to ruin property I. But then
K,
p g
is either a
trefoil knot or a figure eight knot or the trivial knot. It is well known that 1 and —1 surgeries on
the figure eight knot produce the same manifold (the figure eight knot is amphicheiral) whose
fundamental group is the triangle group with presentation
{x,y;x
2
= y =
3
(xy) }
7
and thus
is of infinite order. Therefore the figure eight knot has property I. This completes the whole
proof. •
Corollary 2.4.4 Nontrivial generalized double knots have property P. •
Proof. Similar as the proof of Proposition 2.4.7 and use Corollary 1.1.1, the fact that
(1/2)A&
(1) = pq i- 0 for p / 0 and q ± 0 and Theorem 2.2.1 (ii). O
Chapter 2.
43
On Property I
Corollary 2.4.5
([4]
[26])
Nontrivial double knots have property P. •
Proposition 2.4.7 and Corollary 2.4.4 together give
Corollary 2.4.6
2.4.3
Nontrivial nontrefoil generalized double knots have property PI.
Periodic Knots
In this section we show property I for a few families of periodic knots.
The proof involves
branched covering arguments and applications of results in section 2.2 and section 2.3.
Recall that a knot K in S is called a periodic knot if there is an orientation preserving
3
automorphism / of S with the following properties:
3
1) / has period n > 1, that is, f
n
is the identity map and / ' is not the identity map for
1 < i < n.
2) K is invariant under / , that is, f(K) = K.
3) the fixed point set of / is not empty and is disjoint from K.
Remarks. 1). The action on S by the cychc transformation group {/} generated by /
3
induces a n-fold cychc branched covering p : S —• S /{f}. Due to the positive answer to the
3
3
Smith conjecture [3], the map / is a rotation of S , S /{f} is isomorphic to S , the fixed point
3
set of / is a trivial knot in S
3
3
3
and the image of the fixed point set under p is also a trivial knot
in S .
3
2). The restriction of p on K gives a regular covering p : K —• p(K)
and thus p(K) is also
a knot in p(S )
= 5 . p(K)
Lemma 2.4.5
Let K be a periodic knot in S with period n. If (m,nl) = 1, then S (K,m/l)
3
3
is called a factor knot of K.
3
3
admits a Z action withfixedpoint set a 1-sphere.
n
Proof. Let N be a tubular neighborhood of the factor knot p(K)
in S downstairs disjoint
3
Chapter 2.
44
On Property I
from the fixed point set and let E = S - intN. Then N = p (N)
3
of K in 5
3
is a tubular neighborhood
-1
upstairs and N is invariant under the cychc action of {/}.
Let E = S — intN.
3
Let p, A C dE be a preferred meridian-longitude pair of p(K). Then p ( / j ) C 9/5 is a set of n
-1
copies of meridians and p ( A ) C dE is a preferred longitude of K. Let p. be one of components
- 1
of p~ {p) and X = p ( A ) . Obviously p»[/i] = [/x], p,[A] = n[A] in H\(dE). Let c be a 1-sphere
x
in
_1
with slope m/nl. Then p ( c ) is a set of n copies of 1-spheres in dE with slope m/l.
_1
Attaching n copies of 2-disks to each element of p ( c ) and then filling the n holes with n
-1
3-balls, we extend the cyclic action f\ : dE —• dE onto the solid torus sewn in with the slope
m/l without introducing new fixed points and thus we obtain a Z action on S (K,m/l)
3
n
with
fixed point set a 1-sphere. •
Proposition 2.4.8 Surgery on a periodic knot K in S
3
Proof. B y Lemma 2.4.5, S (K,l/l)
3
admits a Z
n
Suppose that for some slope 1//, S (K,l/l)
3
cannot give a fake Poincare sphere.
action with fixed point set a 1-sphere.
has fundamental group ii2o-
Then
S (K,l/l)
3
is atoroidal by Dehn's lemma. Gordon and Luecke have shown that any homology 3-sphere
obtained by surgery on a knot in 5
3
is irreducible [30]. Now Theorem 2.3.2 implies that
S (K, l/l) is the honest Poincare sphere. •
3
A n immediate consequence of Corollary 2.3.1 and Lemma 2.4.5 is
Proposition 2.4.9 A periodic knot in S
3
with period n ^ 2,3,5 has property I.
So we only need to pay attention to periodic knots with period 2,3 or 5.
Proposition 2.4.10 A periodic knot K with a nontrivial factor knot has property I.
Proof. From the proof of Lemma 2.4.5 we see that S (K,l/l)
3
is a n-fold cychc branched
cover of S (p(K), l/nl) with branch set a 1-sphere. Since niS (p(K), l/nl) is not trivial by
3
3
Chapter 2. On Property I
45
As periodic knots have property P [16], we obtain
Corollary 2.4.7
Example 2.4.1
Periodic knots given in Proposition 2.4-9 and 2-4-10 have property PI. •
Figure 2.9 shows that the knot 8]g is a periodic knot of period 2 with the
figure-eight knot A\ as a factor knot. Hence 8is has property I by Proposition 2.4.10.
Remark. Results preceding this section fail to prove property I for 8is, because
1) the Alexander polynomial of 8
J8
is 13 - 10(i + i
_ 1
) + 5 ( i + r ) - ( i + r ) and hence
2
a
3
3
get A ' ( 8 ) = 1.
18
2) 8is is not a satellite knot. In fact 8js is an alternating knot (i.e. with a knot diagram
where crossings alternate under-over-under- over
as one travels along the knot) and hence
is a hyperbolic knot by ([55] Corollary 1) which asserts that any nontorus alternating knots is
a hyperbolic knot.
46
Chapter 2. On Property I
2.4.4
Strongly Invertible Knots
In this section we investigate property I for strongly invertible knots. The main result of this
section is Proposition 2.4.12 which is a refinement of Lemma 2.2.1 and Lemma 2.2.2 when
specializing to strongly invertible knots. One feature of the argument is that the Kauffman
bracket polynomial, an invariant of links, is used.
A knot K in S is strongly invertible if there is an orientation preserving involution of S
3
3
which carries K onto itself and reverses its orientation.
Note that Waldhausen [79] showed that such an involution is equivalent to a 180°-rotation
of R whose axis meets K in exactly two points.
3
Proofs of the following statements can be found in [15] [5] [57]. Let K be a strongly invertible
knot in S . Then the restriction of the involution to the knot complement can be extended to
3
an involution of the manifold S (K,m/l)
3
S (K,m/l)
3
obtained by performing m//-surgery on K. For each
the quotient under this involution is the 3-sphere S and S (K,m/l)
3
3
is a double
branched cover of S . Moreover the branch set downstairs of this covering can be obtained by
3
removing a trivial tangle from the unknot (i.e. the image of the fixed point set of the original
involution) and replacing it by the m//-rational tangle. In particular if the surgery slope is an
integer m , then the removal and replacement of the trivial tangle corresponding to the surgery
is in fact the attachment of a band with m half twists to the unknot.
By the above discussion, S (K,l/l)
3
admits a
action with fixed point set a 1-sphere.
Hence by the same reasons as given in the proof of Proposition 2.4.8, we obtain
Proposition 2.4.11 Surgery on a strongly invertible knot K cannot give a fake Poincar'e
sphere. •
Proposition 2.4.12 At most one surgery on a strongly invertible knot K can give a manifold
with fundamental group 7i2o-
47
Chapter 2. On Property 1
Proof. By Proposition 2.4.11 and Lemma 2.2.1, we only need to show that S (K, 1) and
3
S (K,-1)
3
cannot both be Poincare spheres. Suppose, on the contrary, that both are Poincare
spheres. By Corollary 2.3.1, there is, up to a conjugation by an isomorphism, a unique involution
on the Poincare sphere with fixed point set a 1-sphere. Hence the associated double branched
covering is the one mentioned in section 2.3 (6). The branched set in the base space S is the
3
(3,5) torus knot up to unoriented automorphisms of S and thus is either the right hand or the
3
left hand (3,5) torus knot.
The branched sets corresponding to S (K, 1) and S (K, - 1 ) , denoted by K\ and K-\, can
3
3
be obtained by band attachments with 1 and - 1 half twist to the unknot respectively. Let U
denote the unknot and let LQ denote the link (of two components) obtained by band attachment
with no twist to the unknot. Then K , K-\, U and LQ have diagrams differing only at the site
x
shown below.
X
We can orient K\, K-\ and LQ in a consistent way such that we can apply the Conway
recursion formula and get V^-j - V / c _ , — zVi
= 0. Since K\ and K.\ are right hand or left
0
hand (3,5) torus knot, it is easy to show, by section 2.2.3 (i) (ii) (iii), that lk(Lo) = 0.
Now we try to get a contradiction by calculating Kauffman bracket polynomials. For unoriented Ki, K-i, U and LQ, we have
{
< Ki > =
< K-i
A <
>=
A '
Lo >
1
<
L
+A-
1
Q
<
> +A <
U >,
U > .
Now consider the oriented K\, K-\, LQ and U (the first three have consistent orientations and
the orientation of U is arbitrarily given). Let w(Lo) = n. Then w(U) = n since lk(Lo) = 0.
Also w{Ki) = n + 1 and tf(A'_i) = n - 1. Hence f {A)
Lo
= (-A)"
3 n
< LQ >, f {A)
v
=
Ch&pter 2. On Property I
(-A)~
3n
< U >=
1, f (A)
Kl
48
=
( - A ) "
^
3
)
1
<
Ki > and / « • _ , ( A )
=
( - A ) ~
3
(
N
-
1
)
<
K-i
>.
Substituting them into (*) above, we have
-A*f (A)
J
=
Kl
(**)<
-A-'f _ (A)
K
Eliminating / L , we get
0
(i) . / R - J =
(ii) . A fx {A)
2
l
A
2
/ K J ( A )
-
A
-
Lo
r
= fL (A) + A
1
2
0
/ * - ^ - ^ )
2
f (A)- A->,
=
A
-
2
A
-
Hence we have either
.
2
= 1 if i f i is ambient isotopic to i f _ i ; or
- A^fKiiA' )
1
= A - A'
2
2
if i f i is the mirror image of K-\.
But both cases contradict the fact that the oriented Kauffman bracket polynomials of right
hand and left hand (3,5) torus knots are / ( A ) = A "
1
6
+ A "
2
4
- A
4
0
and f(A~ )
l
neither of
which fit (i) or (ii). •
For an amphicheiral knot K in S , 5 ( i f , m/l) = S (K,
3
Corollary 2.4.8
3
3
—m/l). Hence we obtain
Amphicheiral strongly invertible knots have property I •
As strongly invertible knots have property P [9], we obtain
Corollary 2.4.0
Amphicheiral strongly invertible knots have property PI. •
Example 2.4.2. The knot 63 is an amphicheiral strongly invertible knot and hence has property
I.
From the discussion in this section we see that basically there is an algorithm for deciding if
a strongly invertible knot K has property I. Namely find the branched knot i n S
3
corresponding
to 1 or —1 suTgery on K and check if the branched knot is a torus knot of type (3,5) or its
mirror image.
2.4.5
Pretzel Knots
In this section we give two infinite families of pretzel knots which have property I.
Ch&pter 2.
49
On Property I
FiguTe 2.10: a pretzel knot of type K(pi,- •••,p )
m
A pretzel knot of type (j>\,P2, • • ••,Pm) in S
Figure 2.10 where each box B
Pi
3
is a knot having a knot diagram as shown in
denotes a two-strand braid with pi half-twists.
First we show
P r o p o s i t i o n 2.4.13 Let K be a pretzel knot of type (p,q,r) such that r is an even number,p +
q
0, p, q are not relative prime. Then K has property PI.
Proof. Since these pretzel knots are strongly invertible, we only need worry about ± 1
surgeries by Proposition 2.4.11 and Lemmas 2.2.1.
A method used by J. Simon in proving property P for such knots (71] can be generalized to
•work for property I as well. Let S be the boundary of a regular neighborhood of the interior of
the obvious nonorientable (since r is even) surface spanned by K. Then 5 is a closed orientable
surface in S , K C S and S- K is connected. Let A, B be the closure of the two complements of
3
S in S .
3
Then both A and B are standard handbodies of genus two. By homological arguments
it can be 6hown that E\(A, S-K)
= Z , Ei(B,
2
S - K) = Z4 where d is the greatest common
divisor of> and q ([71] [72]).
Let N be a tubular neighborhood of K in S
be a meridian and a preferred longitude of K.
3
and let E = S
3
- tni(JV). Let /z,A C dE
Then E\(dE) = Z[p] + Z[X] and [A] = 0 in
50
Chapter 2. On Property I
Let X be a boundary component of an annular neighborhood of K in S.
± 2 ( P + «)[M] in ffiPO =
Then [X] =
and [X] = ± 2 ( p + g)[/x] + [A] in Hx(dX) = Z[/*] + Z[A] ([71] [72]).
We are now going to show that S (K, 1) has infinite fundamental group (the case of —1
3
surgery can be proved in exactly the same way).
It follows from Van Kampen's theorem
that ir\S (K,l) is isomorphic to the free product of *i(A) and iti(B) amalgamated along
3
ITI{S - K) with additional relation X ( a & ) (
± 1
± 2
p +
»)
= 1, where a 6 *\(A\ b 6 JTI(B). By first
: k l
annihilating iri(S—K) and then abelianizing it\(A) and it\(B), we obtain a homomorphism from
TnS (K, 1)onto H^A,S-K)*HiiBiS-K)/
< (ab) ^)
3
±2
>= Z *Z /
±1
2
d
< (ab)* ^^
2
1
>.
By the conditions given in Proposition 2.4.13, ±2(p + q)±l cannot be ± 1 , ± 2 , ± 3 , ± 4 , ± 5 and d
is an odd number. Hence the group Z * Zdj < ( o 6 ) (
±2
2
p+
')
±1
> is not a finite group. Therefore
ic\S (K, 1) cannot be finite. •
3
E x a m p l e 2.4.3 The knot 85 is a pretzet knot of type (3,3,2) and thus has property PI by
Proposition 2.4.13.
Next we point out, as an easy consequence of Proposition 2.4.10,
P r o p o s i t i o n 2.4.14 Pretzel knots of type (2m + 1,2m + 1,2m 4-1), m ^ 0, have property I.
Proof. K(2m + 1,2m + 1,2m + 1) is a knot of period 3 with T ( 2 , 2 m + 1) torus knot as a
factor knot (Figure 2.11). Now apply Proposition 2.4.10. •
Note that W . Ortmeyer showed in [62] that R
3
is the universal cover of each manifold
obtained by nontrivial surgery on pretzel knot of type (4 + 2p, 3 + 2q, — 5 — 2r) with p, q, r
positive. Hence this family of pretzel knots have property PI.
2.4.6
K n o t s u p t o 9 Crossings
Computing A' = ^ A ^ - ( l ) for the classical knots up to 9 crossings, we obtain the following table
of their Casson invariants (we use the knot table given in [65]).
Ch&pter 2. On Property 1
51
Figure 2.11: a pretzel knot of type (2m + 1,2m + l , 2 m + 1) and its factor knot
knot
3i
4,
5,
5
A'
1
-1
3
knot
8i
8
8
A'
-3
0
-4
knot
8»
8l6
A'
4
1
knot
9
A'
0
knot
9
A'
-1
5
knot
9
9 7
9
A'
3
-3
6
8
9
2
9
8
2 2
3 6
9
2 3
3
61
6
-2
-2
1
8
85
8
-3
-1
817
8is
1
7
63
7i
7
1
6
3
5
4
4
1
1
87
8
89
810
811
812
8l3
814
-2
2
2
-2
3
-1
-3
1
0
819
820
821
9i
9
9
9
9
9
1
5
2
0
10
4
9
7
6
7
9io
9n
9l2
9l3
9u
9l5
9l6
9l7
9l8
9l9
9
8
4
1
7
-1
2
6
-2
6
-2
2
924
9s
9
9 7
9
9 9
9o
931
9
9
9
1
0
0
0
-1
-1
-1
2
1
9«
9
9
9
9
0
-2
3
3 8
2
4
2
9
9
3 9
2 6
4 0
-1
2
2
6
2
2 8
4 2
8
2
4 3
-1
2
2
3
0
4 4
2
3
3
4 5
7
9
4
4
3 2
7
5
5
3 3
1
4 6
-2
7
6
6
7
9
7
5
2 0
921
3
3 4
9
-1
7
9 7
9
9
-1
3
4
7
4 8
3 5
4 9
6
This calculation gives immediately that 59 out of the 84 knots have property I (A' ^ ± 1 ) .
But these 59 knots are strongly invertible [36], hence they have property I.
Property I for the knots 4j, 6 , 8s, 8 i has been shown in section 2.4.2, 2.4.4, 2.4.5, 2.4.3
3
respectively. Except for the knots 817, 9
8
3 2
and 9
3 3
, the rest of the knots are strongly invertible
[36]. By the remark given at the end of section 2.4.4 we could decide property 1 for these knots.
52
Chapter 2. On Property I
A l l nontrivial knots with 9 or less crossings have property P since 75 of them have A' ^ 0
and the rest are strongly invertible.
2.5
Concluding Remarks and Open Problems
Let i f be a knot in 5
3
and let E = S — intN(K) be its knot complement. Let F be a closed
3
connected incompressible surface in E. Note that F is necessarily oritentable and it separates
E into two components, say E\ and E , that is, E = E\ U E , E\ fl E
2
Assume that E
2
2
is the component which contains dE.
if there is an annulus A properly embedded in E
2
2
= dE\ fl dE
2
=
F.
The surface F is called a m-surface
with dA consisting of a 1-sphere in F and
a meridian curve in dE. F is called a Im-surjact if there are two disjoint annuli A\ and A
2
properly embedded in E
2
with dA\ — s U m
x
x
and dA
2
= s \J m
2
2
such that si and s are
2
nonisotopic simple closed curves in F and that m j and m2 are meridian curves in dE.
Note
that a m-surface is necessarily nonperipheral and of genus greater than one.
In [55] W . Menasco proved that if K is a knot with a 2m-surface F, then F remains
incompressible in each manifold S (K, m/l) obtained by a nontrivial surgery on K.
3
Hence
knots with 2m-surfaces have property PI by Dehn's lemma.
Q u e s t i o n 2.5.1.
Let K C S
3
be a knot with a m-surface F.
incompressible in each manifold S (K,m/l)
3
Is it true that F remains
with m/l ^ 1/0?
Recall that a Montesinos knot of type (pi/qi,...,p /<7n) is a knot having a knot diagram
n
as shown in Figure 2.12 where each Tp /
i qi
denotes the rational tangle of type pi/qt.
In [60] U . Oertel showed that a Montesinos knot of type ( p i / g i , . . -,p /<ln) with n > 4,
n
qi > 3, i = 1 , . . . , n, is a knot with 2m-surface. Therefore this family of Montesinos knots have
property PI.
Q u e s t i o n 2.5.2. It can be shown that the knots 8 , 6 and 8 1 7 have m-surfaces. Do they have
2m-surfa.ces?
53
Chapter 2. On Property I
Figure 2.12: a Montesinos knot of type ( p i / g \ , p / 9 n )
n
In [74] M . Takahashi proved that no nontrivial surgery on a nontorus 2-bridge knot K can
produce a manifold with cyclic fundamental group. His idea is to show that corresponding to a
nontrivial surgery on K there is a homomorphism from the fundamental group of the resulting
manifold to the group GL(2,C) with noncyclic image.
Q u e s t i o n 2.5.3. For a nontrivial surgery on a nontorus 2-bridge knot, is there a homomorphism
from the fundamental group of the resulting manifold to the group GL(2, C) with infinite image?
Of course the positive answer implies property I for nontorus 2-bridge knots.
Lemma 2.2.1 and Lemma 2.2.2 are quite effective criterions to tell property I for a knot
in S .
3
If there is no fake Poincare sphere, then property I is identical with property I and
things become much simpler by Lemma 2.2.1. For fake Poincare sphere there is also a control
on surgery slopes.
knot in S
3
Recently S. Bleiler and C . Hodgson have shown [6] that if a hyperbolic
admits two finite surgeries then the distance between the two slopes is less than
21(the distance between two slopes m i / / i and m / J
2
2
is | m i / - rn /i|). Hence if 1// surgery
2
2
on a hyperbolic knot produces a fake Poincare sphere, then \l\ < 21. To further eliminate the
possibilities of obtaining fake Poincare sphere by surgery on a knot in S , i l might be helpful
3
to consider the approach suggested by the following two questions.
Q u e s t i o n 2.5.4. If S (K, 1 //) is a fake Poincare sphere, is it homotopy equivalent to the honest
3
Poincare sphere?
Chapter 2. On Property I
54
Question 2.5.5. Is the Casson invariant a homotopy type invariant?
From the discussion in section 2.4.3, we see that to solve property I for periodic knots, it is
equivalent to solve the following
Problem 2.5.1. Let A" be a periodic knot with period 2 or 3 or 5 and with a trivial factor
knot p(K). Determine when the branch set, a trivial knot, i n S downstairs of the covering
3
p : S —•* S /{f) becomes a torus knot of type ( ± 3 , 5 ) or ( ± 2 , 5 ) or ( ± 2 , 3 ) respectively after
3
3
performing 1 or —1 surgery on p(K) in S .
3
From the discussion in section 2.4.4, we see that to solve property I for strongly invertible
knots in 5 , it is enough to solve the following
3
Problem 2.5.2 Determine precisely when a trivial knot can be changed to a torus knot of type
(3,5) or (—3,5) by a band attachment with a half twist to the trivial knot.
We may also raise
Problem 2.5.3. Solve property I for the knots 817, 932 and 933.
Problem 2.5.4. Solve property I for symmetric knots.
Question 2.5.6. Is there a nonsymmetric knot K C S such that some nontrivial surgery on
3
K gives a manifold with finite fundamental group?
Chapter 3
O n B o u n d a r y Slopes
3.1
Introduction
Let K in S
3
S
3
be a nontrivial knot, let N(K)
be a tubular neighborhood of K in S , and let E —
3
— intN(K) with the preferred meridian-longitude framing pair on dE.
If (F, dF) C (E,
dE)
is an orientable, incompressible and ^-incompressible surface (with dF nonempty), then the
components of dF all have the same slope on dE and such a slope is called a boundary slope.
Consider (p(K)
C Q U {1/0}, the set of boundary slopes of K.
Questions about (f(K)
are
closely related to understanding the 3-manifolds obtained by Dehn surgery on K (very possibly
a Haken manifold is produced by surgery with a boundary slope [16]). In [33] A . Hatcher
and W . Thurston completely described <p(K) for 2-bridge knots. In particular they found that
<p{K) C Z U {1/0}
for every 2-bridge knot. The following natural question was thus raised in
[33].
Q u e s t i o n . ([33]) Is it true that <p(K) C Z U {1/0}
for every knot K in S
3
?
In this chapter we give the question a negative answer by showing that for the (—2,3,7)
pretzel knot there exists a nonintegral boundary slope. The proof is given in the next two
sections. In Section 3.2 we prove
T h e o r e m 3.1.1 If K C S
3
is hyperbolic and not sufficiently large and if K admits two non-
trivial cyclic surgeries, then there exists at least one nonintegral boundary slope for K.
The set of knots satisfying the conditions given in the above theorem is not empty. In
55
Chapter 3.
On Boundary Slopes
56
Section 3.3 we prove
L e m m a 3.1.1 The pretzel knot of type (—2,3,7) ts hyperbolic and not sufficiently large and
admits Z\& and Z\$ surgeries.
Section 3.4 concludes with remarks and open problems.
This chapter was essentially contained in the author's paper [84]. Infinitely many nonintegral
boundary slopes have been found by A . Hatcher and U . Oertel by a different approach [34].
3.2
P r o o f o f T h e o r e m 3.1.1
We apply main results of [16].
By Theorem 1.1.3, the two nontrivial cychc surgery slopes that K admits are successive
integers, say, m and m + 1.
Claim. Neither m nor m + 1 is a boundary slope.
Proof of Claim.
Suppose that one of the two slopes, say m , is a boundary slope. Let
(F, dF) C (E, dE) be an orientable essential surface such that dF is a nonempty set of boundary
curves in dE of slope m and such that the number of components of dF is minimal subject
to these conditions. Note that in any knot complement all orientable essential surfaces except
those with 0 boundary slope are separable surfaces. Now applying [16] Proposition 2.2.1 if F is
nonplanar or applying [16] Proposition 2.3.1 if F is planar, we arrive at a contradiction either
with the condition that irxS (K,m)
3
is cychc or with the condition that K is not sufficiently
large. •
Since i f is a hyperbolic knot, the interior of E has a complete hyperbolic metric of finite
volume. We can now apply results of [16] Chapter 1. It follows that there exists a norm || • ||
on the 2 dimensional real vector space Hi(dE,R)
(1).
such that
|| • || is positive integer valued for each (m,Z) £ Hi(dE,Z)
- {(0,0)} C
Hi(dE,R).
Chapter 3.
On Boundary Slopes
57
Note that every slope m/l G Q U {1/0} is corresponding to the pair of primitive elements
(±m,±/)€
H {dE,Z).
x
(2) . Define n = min{\\(m,l)\\;(m,l) G Hi(dE,Z)-(0,0)}
n i n Hi(dE,R).
and consider the ball J5 of radius
Then i? is a compact, convex, finite sided polygon which is symmetric about
the origin (i.e. -B = B). Note that intB fl H (dE,Z)
x
= (0,0).
(3) . For any vertex of B, there is a primitive element (m,/) G H (dE, Z) such that (m,/)
x
lies on the semi-line starting at (0,0) and passing through the vertex and moreover m/l is a
boundary slope.
(4) . If m/l is not a boundary slope and S (K,m/l)
3
has cyclic fundamental group, then
( ± m , ± / ) G dB (of course they are not vertices of B by (3)).
(5) . Assume that the area of a parallelogram spanned by any pair of generators of H (dE, Z)
x
is 1. Then AreaB < 4.
Now to prove Theorem 3.1.1 it suffices to show that there exists a vertex of B which provides
a nonintegral boundary slope i n the way described in (3). B y the Claim and (4) above, points
( ± m , ±1) and ( ± m ± 1, ±1) are all on the boundary of B and none of them are vertices of B.
Let T be the closed edge segment of dB on which point (m + 1,1) lies ( as an interior point)
and let v — (si,s ) and v = ( i i , t ) be the two vertices of T. Let L be the line i n H\(dE,R)
x
2
2
2
passing through points {(m, 1); m G Z}.
C a s e 1. T is not parallel to L.
Then one of the vertices of T, say v = (si,s ) must lie above the line L i n the sense that
x
2
s > 1. Such a vertex certainly determines a nonintegral boundary slope i n the way described
2
in (3).
C a s e 2. T is parallel to L.
Then m G T (as an interior point) and v = ( « i , l ) , v = ( r j , l ) . We may assume that
x
2
si < m < m + 1 < t . We must have m — l < s i < * i < m + 2 since otherwise the area of
x
Chapter 3.
On Boundary Slopes
58
B would be large than 4, violating (5). Now both v\ and v determine nonintegral boundary
2
slopes as required. •
3.3
P r o o f o f L e m m a 3.1.1
Throughout this section let K denote the pretzel knot of type (—2,3,7). Fintushel and Stern
have shown (unpublished)
L e m m a 3.3.1 ( R . F i n t u s h e l a n d R . Stern) 18 and 19 Dehn surgery on K yield lens spaces.
For the sake of the completeness of the paper we give the following verification of their
result.
Proof. The idea is to show that 18 and 19 surgeries on K yield manifolds that double
branched cover 5
3
with branched set in S
3
a 2-bridge link and a 2-bridge knot respectively.
The manifolds are therefore lens spaces. Actually we will see that they are £ ( 1 8 , 5 ) and £ ( 1 9 , 8 ) .
We provide below an explicit pictorial illustration.
Note that i f is a strongly invertible knot (Figure 3.13 (a)). The quotient under the involution
shown in Figure 3.13 (a) is S
3
and hence S
3
double branched covers S
3
with branched set
downstairs the unknot as shown in Figure 3.13 (d) (the process is shown through Figure 3.13
(a)-(d)).
As noted in section 2.4.4, the strong inversion on K can be extended to an involution
on each of the manifolds S (K,m/l)
and the quotient under the corresponding involution is
3
S . Moreover the branched set in S of the corresponding double covering can be obtained by
3
3
removing the trivial 1/0-tangle (ball B shown in Figure 3.13 (d)) from the unknot and replacing
it by the rational m//-tangle (beware that the sign of a rational tangle given here is opposite to
that given in [15]). In particular the branched sets in S corresponding to 18 and 19 surgeries
3
are shown in Figure 3.14 (a) and (b) respectively. They turn out to be (by isotopy) the 18/5
Chapter 3. On boundary Slopes
59
Chapter 3.
60
On boundary Slopes
isocopy
21
preferred
laccitude
Figure 3.13: surgery on (-2,3,7) pretzel knot and double branched covering
Chapter 3.
On Boundary Slopes
61
18/5
2-bridge link
(B) branch set corresponding to 1 8 - s u r g e r y
19/8
2-bridge knot
(b)
branch set corresponding to 1 9 - s u r g e r y
Figure 3.14: branched sets of 18- 19-surgeries on the ( - 2 , 3 , 7 ) pretzel knot
2-bridge link and the 19/8 2-bridge knot. Therefore the manifolds upstairs are lens spaces
1 ( 1 8 , 5 ) and 1 ( 1 9 , 8 ) .
•
Reference for the argument above is [5] [15] [57] and [65].
Lemma 3.3.2 Ji is hyperbolic and not sufficiently large.
Proof. Note that K is the K{-l/2,1/3,1/7)
star knot (notation as in [60]) and hence by
[60] Corollary 4 (a), K is not sufficiently large. K cannot be a torus knot either since there is
no nontrivial torus knot which could admit 18 and 19 cychc surgeries by Theorem 1.1.1. •
Lemma 3.1.1 follows Lemma 3.3.1 and Lemma 3.3.2. O
Chapter 3.
3.4
62
On Boundary Slopes
P r o p e r t i e s o f <p(K) a n d O p e n P r o b l e m s
In this section we list known properties of f(K) and point out some open problems.
T h e o r e m 3.4.1 ([17]) \<p(K)\ > 2 for any nontrivial knot K in S .
3
Theorem 3.4.1 is sharp as a torus knot T(p, q) has exactly two boundary slopes, namely
<p(T(p,q))={0,pq}.
Q u e s t i o n 3.4.1. Is it true that for a nontorus knot K in 5 , Iv^-SQI > 2?
3
T h e o r e m 3.4.2 ([32]) <p(K) is afiniteset for any knot in S .
3
In spite of Theorem 3.4.2, there is no up bound restriction on the distance among boundary
slopes in <p{K) when K varies over all knot types.
This is easily seen to be true when K
varies in the set of cabled knots of a fixed knot, namely the distance between the boundary
slopes 0 (0 E <p(K) for all knots K C S ) and pq (the slope of the cabling annulus) can be
3
arbitrarily large.
This is also true when K varies over the set of hyperbolic knots. In fact,
by Examples 1.4.1, Fintushel-Stern knots K
2n
are hyperbolic knots admitting cychc surgeries.
Then a similar argument as that given in Theorem 3.1.1 will give a boundary slope m/l of K
2n
with |m| > |18n|. Recently A . Hatcher and U . Oertel investigated <p(K) for Montesinos knots
and they found infinitely many Montesinos knots having nonintegral boundary slopes. By their
results \l\,m/l £ <p(K) has no universal bound when K varies over knot types.
In the proof of Theorem 3.1.1 one of properties of the fundamental domain B is that each
vertex of B corresponds to a boundary slope. Let m/l
0, be a boundary slope of a hyperbolic
knot K in S and let L C H\{dM, R) be the semi-line which starts from (0,0) and passes (m, /).
3
Q u e s t i o n 3.4.2. Does L intersect £ at a vertex of Bl
If the answer is yes, then some interesting information about cychc surgery and boundary
slopes can be obtained. In particular Theorem 3.4.2 follows for hyperbolic knots.
Chapter 3.
On Boundary Slopes
63
Question 3.4.3. Is Theorem 3.1.1 still true if in Theorem 3.1.1, the condition ' i f admits two
nontrivial cyclic surgeries' is reduced to ' i f admits one nontrivial cyclic surgery'?
If the answer is yes, then all if2 (M > 1) have nonintegral boundary slopes.
n
Example 1.4.2 shows that each Berge-Gabai knots J
n
admits two nontrivial integral surg-
eries.
Question 3.4.4. Is J
n
not sufficiently large?
If the answer is no, then J
n
has at least one nonintegral boundary slope by Theorem 3.1.1.
Let p<p(K) be the set of boundary slopes of essential planar surfaces i n S — intN(K).
3
Theorem 3.4.3 ([29]) \p<p(K)| < 6 for any knot i f .
Theorem 3.4.4 ([31]) p<p(K) C Z U {1/0} for any knot i f .
It is well known that for any torus knot i f = T(p,q) or any cabled knot i f — C(p,q),
pq G p(p(K). It is also known that for certain prime knots, e.g., those which have prime tangle
decompositions [48], and even for certain hyperbolic knots, e.g. those which have simple tangle
decompositions [73], 1/0 G pip(K) (the proof is not too hard and is omitted). However
Question 3.4.5. Is it true that pip(K) — {1/0} = 0 for any nontorus noncabled knot Kl
Note that a positive answer to this question proves the cabling conjecture which states: i f
S (K, m/l) is a reducible manifold then i f is a torus knot or a cabled knot.
3
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