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Topics on Dehn surgery Zhang, Xingru 1991

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TOPICS ON D E H N SURGERY By Xingru Zhang B.Sc. of Mathematics, Nanjing Institute of Posts and Telecommunications A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES MATHEMATICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA January 1991 © Xingru Zhang, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M.flfjt.l^ ri(3t f The University of British Columbia Vancouver, Canada Date A -^.| tf?i DE-6 (2788) Abstract Cyclic surgery on satellite knots in S3 is classified and a necessary condition is given for a knot i n S3 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 in S3 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. i i Table of Contents Abstract " List of Figures v Acknowledgements V 1 Introduction v " 1 On Cyclic 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 2 On Property I 18 2.1 Introduction 18 2.2 Prehminaries 20 2.2.1 The Casson Invariant and Property I (I) 20 iii 2.2.2 The Rohlin Invariant and the Arf Invariant 25 2.2.3 The Conway Polynomial and the Kauffman Bracket Polynomial 26 2.3 Cyclic Actions on the Poincare Homology 3-Sphere 28 2.4 Knots Having Property I or I 35 2.4.1 Torus knots, Slice Knots and Knots Formed by Band Connect Sums . . . 35 2.4.2 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 2.5 Concluding Remarks and Open Problems 52 3 On Boundary Slopes 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 Kn 13 1.2 Berge-Gabai knots Jn 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#bK2 is r-equivalent to K1^K2 39 2.7 a generalized double knot 41 2.8 generalized twisted knot KVA 42 2.9 8i8 has 4i as a factor knot 45 2.10 a pretzel knot of type K(pi, • • • ,pm) 49 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 n / o n ) 53 3.13 surgery on (—2,3, 7) pretzel knot and double branched covering 59 3.14 branched sets of 18- 19-surgeries on the (—2,3,7) pretzel knot 61 v Acknowledgements I wish to express my gratitude to my supervisor, Professor Erhard Luft , for his invaluable guidance, encouragement and support. I also would hke to thank the University of British Columbia for its generous financial assistance. F ina l thanks go to my family, especially to my wife, Li juan Zhang, for their emotional support. vi I n t r o d u c t i o n One of the basic methods to construct closed orientable 3-manifolds is by Dehn surgery on knots or links i n the 3-sphere S3, 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 in 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 S3 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 3 , it is in general not known which manifold can be obtained by which 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 S3 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 i f 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 in 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 in the sense described above are of considerable interest. In this paper we address three topics concerning Dehn surgery along this line. vii Topic 1. Which Dehn surgery on which knot in S3 can yield a lens space? More generally which Dehn surgery on which knot in S3 can yield a manifold with cychc fundamental group? Topic 2. Which Dehn surgery on which knot in 5 3 can yield the Poincare homology 3-sphere? More generally which Dehn surgery on which knot i n S3 can yield a manifold with 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 in S3 is classified and a necessary condition is given for a knot i n S3 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 S3 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. v i i i 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) of K in M and sew it back in by an isomorphism of tori. Let E = M — intN(K) 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} where m and / are integers with (m,l) = 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)) = Z[X] and [A] = 0 in H\(E) — 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\m\- Hence M(K,m/l) is a homology 3-sphere iff |m| = 1. Let S3(K, m/l) denote the closed orientable 3-manifold obtained by surgery with slope m/l 1 Chapter 1. On Cyclic Surgery 2 along a knot K in S3. If S3(K,m/l) is a manifold with cychc fundamental group (if so the group is Z|m|), then the corresponding m//-surgery is called a cyclic surgery or a Z|m| surgery. In particular if S3(K, m/l) is a lens space, then the corresponding m//-surgery is also called a 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 3 then surgeries on O produce all lens spaces (including S3 and S2 x S1) and S 3(0, m/l) = l (m, /). 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 is the lens space L(m,nq2). 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,4g2). 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 S3 admits a cyclic surgery, then the surgery slope 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 S3 gives a manifold 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,4q2). 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 S3 have property P. Therefore classification of cychc surgery on knots in S3 reduces to hyperbolic knots. There 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 S3 admits a symmetry (i.e. is invariant under a finite group action on S3) which is not a strong inversion (see section 2.4.4 for the definition), then there exists no 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 S3 never give a manifold with infinite cychc 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. Chapter 1. On Cyclic Surgery 4 Our second result in this chapter gives a necessary condition for a knot in S3 to admit a 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 : Mm/i —• 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)) = Zm) and let q : M(m) —• M be the \m\-fold cyclic branched regular covering of M with branch set K in M. Then Mm/i is a homology 3-sphere iff M(m) is a homology 3-sphere. Corollary 1.1.2 If a knot K in S3 admits a nontrivial cyclic surgery with slope m/l, \m\ > 1, then the \m\-fold cyclic branched cover of S3 branched over K is a homology 3-sphere and thus lni=im ' Ax(e2, r ,J'/lml)| = 1 where AK(t) is the Alexander polynomial of K (see [19]). • It was shown by S. Bleiler and R. Litherland in [7] that the projective space RP3 cannot be obtained by surgery on any nontrivial symmetric knot in S3. This result has been generalized by S. Wang and Q. Zhou in [81] to : No nontrivial symmetric knot in S3 admits /^ -surgery. As a special case of Corollary 1.1.2 we have that if some knot K in S3 admits Z2-surgery, then 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] (IO124 is the (3,5) torus knot), hence we have Corollary 1.1.3 Surgery on any nontrivial knot of 10 or less crossings cannot give a manifold M with irx(M) = Z 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 S3 nontrivially (i.e. not isotopic 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 S3 be a satellite knot and let K* be a nontrival companion knot of K. Let N* = K* x D2 C S3 be a solid torus neighborhood of K* in S3 with K C intN* and let E" = S3 - intN*. 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 orienta-tion for K. Then w > 0 is the winding number of K in N*. Let N = K x D2 C tniiV* be a solid torus neighborhood of K in N* and let E = S3 - intN and EQ — N* — intN. Let p, A be a preferred meridian-longitude pair of dN = dE, that is, HiidN) = H\(dE) = Z[p] + Z[X], u = 0 in Hi(N) = Z[X] and [A] = 0 in HX(E) = Z[u]. Then tfi^o) = Z\p](BZ[\*]t [A] = u[X*] in H^EQ) and [fi*] = u[p] in # i ( £ 0 ) (by choosing proper orientation for p and A). Let N*(K,m/l) denote the manifold obtained by surgery along K in N*. As S3(K,m/l) = Chapter 1. On Cyclic Surgery 6 E*UN*(K, m/l), we may obtain some information about S3(K, m/l) by first considering surgery in N* along K. The following lemma proved in [28] by C M . Gordon gives precise homological information about N*(K,m/l). Lemma 1.2.1 ([28]) Lemma 3.3) (i). Hi(Nm(K,m/l))* Z© Z(WyTn). (ii). ker(Hi(6N*(K, m/l)) —> HriN^K, m/l))) is the cyclic subgroup ofHi(dN'{K, m/l)) generated by [/*•], ifu = 0. 1.2.2 D . Gabai's Results Theorem 1.2.1 and Theorem 1.2.2 below are main results from [22] and [25] proved by D. Gabai. These theorems are not only applied in this chapter hut also in chapter 2. Recall that a knot in a solid torus is called an n-bridge braid if the knot can be isotoped in the solid torus to a braid which lies in the boundary of the solid torus except for n bridges. We first restate D. Gabai's main result in [22] as follows with more information added in case 2) due to M. Scharlemann [67]. Theorem 1.2.1 ([22]) Let K be a knot in a solid torus N* with nonzero wrapping number. Perform m/l-surgery along K in N* and let K' be the core of the sewn solid torus. Then one of the following must hold: 1) . N*(K,m/l) is a solid torus. In this case both K and K' are 0 or 1-bridge braids in N* and Nm(K, m/l) respectively. 2) . N*(K,m/l) = D2 x 51#I(s,r) where L(s,r) is a nontrivial lens space (\s\ > \), K is a cabled knot and m/l = rs. 3) . N*(K,m/l) is irreducible and dN*(K,m/l) is incompressible. Chapter 1. On Cyclic Surgery 7 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)). By [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 Px = PC]E0. Then (Pi,dPi) C (E0,dE0) is a planar surface whose u> (winding number of K) inner boundaries (fat vertices), P\ f)dN(K), all have the same orientation induced from Pi and K. Note that the inner boundaries of P\ are meridians of Chapter 1. On Cyclic Surgery 8 K and P\ has only one boundary component on dN*. Analogously, there is a planar sur-face (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) = D2 x S^ftL^^). 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 number 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. On Cyclic Surgery 9 1.3 Proof of Theorem 1.1.4 By Theorem 1.1.3, we may assume that / = 1 Lemma 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, m) is reducible. Then by Theorem 1.2.1. 2), K is a cabled knot C(r,s) on K* with \s\ > 1 and m = rs. By [28] Corollary 7.3, S3(K,rs) * S3(K\r/s)#L(s,r). Hence *i(S*(K,rs)) S *i(S3(Km, r / s ) ) *7r 1 ( I (5 , r)). If K* is a torus knot, then Ttx(S\K*, r/s)) ± 1 by Theorem 1.1.1; if K* is a nontorus knot, then by Theorem 1.1.3, wi(S*(Ktr/s)) ^ 1. Hence ni(S*(K, m)) is a free product of two nontrivial groups, contradicting the assumption that 7 T i ( 5 3 ( i i ' , m)) is cychc. Hence N*(K,m) is irreducible. Since iri(S*(K,m)) is cychc, dN*(K,m) is a compressible torus in SS(K,m). Let B2 C S3(K,m) be a compressing 2-cell for dN*(K,m). Since K* is nontrivial, B2 C N*(K,m). Performing 2-surgery on dN*(K, m) using 5 2 , we get a 2-sphere which must bound a 3-ball in 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 B2 be a proper meridian 2-cell of N*(K, m). Then [dB2] is a primitive element of HxidN'iK^m)) and [dB2] e keriH^dN'iK,™)) —* Hi(N*(K,m))). By Lemma 1.2.1 (ii), in . H i ( £ # • ( # , m)). Hence Chapter 1. On Cyclic Surgery 10 Since w ^  0, S3(K,m)=S3(K\^) = S3i (if*, and thus m Z^ = H1(S\K>m))=H1(S\K\ )) = Z |m| . Hence (w2,m) = 1. 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>2 = 1 and thus 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(53(A',m)) = 7ri(53(if*,m/u>2)) can possibly be cychc only when m is equal to (*) u2pq±l. 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 \u2pq + 1 ± (t + JU>)OJ ± b\ > 0, \u2pq - 1 ± (t + ± b\ > 0, \u2pq + 1 ± (z + ± 6 ± 1| > 0 and \u2pq - 1 ± (t + ± b ± 1| > 0. We verify the first inequality. The rest of inequalities can be similarly verified. Chapter 1. On Cyclic Surgery 11 If \pq±j\ ± O.then \u2pq+l±(t+ju)u±b\ = \(pq±j)u2±tu±b+l\ > \pq±j\u;2-tu-b-l > u2 - (u - 2)w - (w - 2) - 1 = u> + 1 > 0; If \pq ± j\ = 0, then \u2pq + 1 ± (t + ± b\ = | ± tu ± b + 1| > tu> + b - 1 > 0. • 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 3 and let N = q~l(N). Then E - M(m) -tntJV E = M — intN is the m-fold cychc regular unbranched covering associated with the kernel of the composition iti{E) —• Hi(E) = Z —> Z m . Let p,, X C BE = ON be a preferred meridian-longitude pair. Then fi = q_1(p) is a meridian curve ofdE = dN. 9_1(A) C dE = dN 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). Then we may assume M(K,m/l) - intN' = E. Since p, C dN' is a generator of Hx{M(K,m/l)) = Zm, p-\N') is a solid torus. Let JV' = and let E' = Mm/l - intN'. Claim. (p\U*i(E')) = n(E). Proof of Claim. Let a e *i(E'). Then p»(a) = 0 in Hi(M(K,m/l)) = Zm by the definition of p : Mm/l M{K,m/l). Therefore (p|).(>i(.E')) C ker(*i(E) —•* i7x(J5) = Z —• Z r a) = 7ri(J3). On the other hand both (p|)*(7r1(J5')) and 7 r 1 ( £ ' ) are normal subgroups of it\(E) of index m. Hence (p|)»(7Ti(J3')) = 7Ti(i5). • 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]. Chapter 1. On Cyclic Surgery 12 Then p _ 1 (M*) consists of m disjoint 1-spheres p.*, j = l , . . . , m . \fim] = [p.] + l[X] in iri(dE) = HiidE) = Z[fi] + Z[\] since = (PI).([M] + TO in Tr ^ d E ) = H^OE). Hence [A] = hx*] in H\(E). If M(m) (Mm/i) is a homology sphere, then E\{E) = Z and [/2] = [p") is a generator of R~i(E). Consequently Mm/i (Mm) is a homology 3-sphere. • Remark. As pointed out earlier, Corollary 1.1.2 gives, in particular, the result that if a knot in S3 admits Z2-surgery, then the determinant of the knot is 1 or - 1 . The following proposition which is a consequence of a result of [10] also provides a necessary condition for a knot in S 3 to admit a i?P3-surgery. Proposition 1.4.1 Let K be a knot in S3 and let Ax(t) be the normalized Alexander polyno-mial of K, i.e. AK(l) = 1, Aif(r') = AK(t). If K admits RP3-surgery, then A £ ( l ) = 0 where " denotes the second derivative. Proof. A surgery formula for calculating the generalized Casson invariant, as defined in [10], of the oriented manifold S3(K,m/l) is given in [10], namely \(S3(K,m/l)) = ( / / 2 m ) A £ ( l ) - (sgn(m)/2)s(l, m), where s(Z, m) = TJ^iJlm - [j/m] - l/2)(;7/m - [jl/m] - 1/2) is the Dedekind sum of / and m. As RP3 can be obtained by 2-surgery on the trivial knot O, it follows that X(RP3) = ± A ( 5 3 ( 0 , 2 ) ) = -5(1,2) = 0. Now suppose that S 3 ( A ' , ± 2 ) = RP3. Then 0 = A ( 5 3 ( A ' , ± 2 ) ) = ( ± 1 / 4 ) A £ ( 1 ) . • 1.5 Examples, Remarks and Open Problems Example 1.5.1. Fintushel-Stern knots Kn. R. Fintushel and R. Stern [21] showed, using the Kirby-Rolfsen calculus, that 9n surgery on the knot Kn 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 Kn of S 3 branched over K-^n can be obtained by -1/n surgery on the figure eight knot. By W. Thurston's work [77] the cover is a hyperbolic non-Haken manifold. Hence the knot K7n is 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 K2 also yields a lens space (see section 3.3 for an amusing verification of this result). Question 1.5.1. Is there any other Kn, \n\ > 2, which admits two nontrivial successive integral cyclic surgeries? Example 1.5.2. Berge-Gabai knots Jn.° 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 Jn Embedding V into S3 as a trivial solid torus, we obtain infinitely many knots Jn (Figure 1.2) in S3 such that each of Jn admits two nontrivial successive integral surgeries with slopes -30 + 49n and -31 4- 49n. Note that every Jn is a hyperbolic knot by Theorem 1.1.1 and Theorem 1.1.4. Recall that a knot K in S3 has free period n if there is a periodic transformation T of S3 with order n such that {T} acts on S3 fixed point freely and leaves Ii setwise invariant. In [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 3 admits a nontrivial lens space surgery with slope m/l, |m| > 1, then there is a knot K' C S 3 of free period \m\ such 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 S3 of free periods. Chapter 1. On Cyclic Surgery 15 Proof. .Since there are infinitely many hyperbolic knots admitting lens space surgery (Ex-amples 1.5.1-2), the knot exterior of each of these knots is covered by a knot exterior of a knot in S3 with free period. Each of these free periodic knots is hyperbolic. This follows from the 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 if, in S3, then K' is a torus knot or a hyperbolic knot or a satellite 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 if 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-1(T) is a set of essential tori 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 Chapter 1. On Cyclic Surgery 16 Corollary 1.5.1 There are infinitely many hyperbolic knots in S3 whose knot groups can be imbedded into knot groups of hyperbolic knots as normal subgroups with finite cyclic quotients. • 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 S3 whose exterior 5-sheeted covers the exterior of T(2,3) and the 5-sheeted cover of S3 branched over T(2,3) is a homology 3-sphere Q. Actually K' is the left hand trefoil 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 S3 branched over T(2,3) is the 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 S3 branched over T(2,3) is the Seifert fiberred manifold 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 Kn and Jn. The following conjecture was raised in [81]. Cyclic Surgery Conjecture. ([81]) For a nontrivial knot K in S3 and a nontrivial slope m/l, \irtS3(K, m/l)\>4. 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 in S3 admits a nontrivial cychc surgery of integral slope m. If i f can be isotoped nontrivially into a sohd torus V in S3 (i.e. K is not isotopic to the core of V 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 in S3. If i f is a 1-bridge braid, then by Theorem 1.2.2 and by presentation of 1-bridge braid in a solid torus, it can be shown that |m| > 4. Question 1.5.3. Let i f C S3 be a hyperbolic knot which admits a nontrivial cychc surgery with slope m. Can i f be isotoped nontrivially into a solid torus V in 5 3 such that m-surgery on V along i f yields a solid torus again? (all known knots in S3 that admit cychc surgery have 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 S3. Examples of homology 3-spheres which can be obtained by surgery on two or finitely many distinct knots in S 3 have been given [47] [65] [13] [52] [53]. 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 2 o- We call a 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 3 has property I if every surgery along K does not yield a manifold M with 7Ti (M) = ii2o- A knot K in S3 has property I if every surgery along i f does not yield the Poincare sphere. Of course the trefoil knot does not satisfy property I. Conjecture I (I). Every nontrefoil knot in S3 has property I (I). 18 Chapter 2. On Property I 19 Recall that the property P (P) conjecture states that every nontrivial surgery along a non-trivial 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. Conjecture 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 fol-lowing 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 Arf 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. On Property I 20 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\ = dV2 = F 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: V i / \ F* —>• F M \ / V2 induces the following diagram of surjections on their fundamental groups: / \ H\F* • 7 r XF TTiAf. \ / * i V 2 Chapter 2. On Property I 21 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 in turn induces the following diagram of injections on representation spaces: s \ \ / R(*iV2) Let R(TTiF)red C R(ir\F) be the set of reducible representations, i.e. the set of homomor-phisms from iriF to SU(2,C) with abelian images. Similarly define R{TC\Vi)Ted and R(-K\M)red-Let Q = image of RfaM) in R* = RfaF*), Qi — image of R(iriVi) in R*, R = image of R(it\F) in R", A =image of R(ir\M)Ted in R*, Ai = image of R(^iVi)red in R*, B = image of R(it\F)Ted in i i * . 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 Q2 can be moved by an isotopy in R to Q2 such that Q i and Q2 intersect transversally at finitely Chapter 2. On Property I 22 many points in R. The orientation of M can be used to determine an orientation of Qi, R", Qi and R. Therefore an algebraic intersection number < Q\,Q2 >fi=< Q11Q2 >ft c a n D e denned. Also note that R* is a manifold isomorphic to (S 3 ) * 5 and Qi C R*, i = 1,2, are compact submanifolds of middle dimension, both being isomorphic to (S3)3. Let < Qi,Q2 >«• be the homological intersection number of Q\ and Q2 in R*. Then Casson invariant of M, denoted by A, is given by A . Casson proves that this number is an integer and is independent of the Heegard decom-position of M. Note that | < Q\,Q2 >R* \ - \R\(M)\ = 1, therefore < Qi,Q2 >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 ( A . Casson) Let K be a knot in an oriented homlogy 3-sphere M and let M(K,\[l) be the homology 3-sphere obtained from M by performing 1/l-surgery on K. Let AA-(t) be the normalized Alexander polynomial of K, i.e. A j f ( l ) = 1 and A j c ( t - 1 ) = Atf ( f ) . 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 3 we shall call X'(K) = (1/2)A'£(1) the Casson invariant of K. Chapter 2. On Property I 23 Let T denote the right hand trefoil knot and D3 the Poincare homology sphere. Since D3 can be obtained by 1-surgery along T and AT(t) = - 1 +1 + r " 1 , we have A(I>3) = ( 1 / 2 ) A £ ( 1 ) = 1. Now suppose that S3(K,l/l) is the Poincare 3-sphere obtained by 1/i-surgery along a knot K in S3. Then by Theorem 2.2.2, X(S3{K,l/l)) = / ( 1 / 2 ) A £ ( 1 ) = 1 or - 1 . It is known that the normahzed Alexander polynomial of any knot K in S3 can be expressed as A.R-(*) = a 0 + Ei=i + *"*') € Z[t,t~l). By a simple calculation we get ( 1 / 2 ) A £ ( 1 ) = £ J = J ai{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 S3. IfS3(K, l/l) is the Poincare sphere, then X(S3(K/l/l) = 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;x2 = (xy)3 = ys,x4 = 1}. Let p : 7i2o —> SU(2,C) be an irreducible representation. Note then p(/i 2o) must be a non-abelian subgroup of 517(2, C). ( 1). Claim. p{x)2 = , 1 0 , Proof of Claim. (p(x)2)2 = p(x)A = p(x4) = | |. Thus the eigenvalues of p(x)2 v 0 1 1 0 \ are either 1 or - 1 . Consequently p(x)2 — j | or p(x)2 = 0 1 1 0 . , then p(x) 0 1 P(y)5 = o 1 / ] or p(x) = j |. Therefore p(xy)3 = ±p(y)3 = 0 1 / V 0 - 1 / and thus p(y) = p(Zi2o) is non-abelian. 1 0 . But this contradicts the assumption that 0 1 Chapter 2. On Property I 24 2). Note that 0 1 -1 0 A 0 0 A"1 0 -1 1 0 L - l 0 A Since p(x)2 = p(xy) 3 _ , -1 0 \ I i 0 p(y)h = | I, it follows that p(x) is conjugate to | |, p(y) is conjugates to 0 -1 / I 0 -» e? 0 0 e & 0 , or | j and p(xy) is conjugates to 0 e~*f~ / 0 e-""'/3 In particular the trace of p(xy) is tr(p(x)p(y)) = e " / 3 + e - "* / 3 = 1. 0 3). After a conjugation, we may assume that p(y) = I V e s , n = 1 or 3. Then \ 0 e~~ P(x) e {B ° IB^-B 6 SU{2,C)} = {[ ** ^ | ;ie72,5eC,t 2 + |&|2 = l}. 0 -i j \ - 6 -rz From 2) we conclude that 1 = tr(p(x)p(y)) = tie^ — tie "s^" = —2tsin~- and thus t = - 1 2*tn I 5 2 - and |6|2 = 1 — t 2 = 1 — . • \ „ , . Let c — c(n) = 1 — . As c > 0, |6|2 = c has solution set b = C 2 e e « , 0 G [0,27r). Hence we may further assume that / {p(x),p(y)} = { PI _ c T 0 0 e~ -cke-0i h~6i C2e 2sm£ ~2^f 0 2 e s 0 0 e~~ nm e s 2st'n^ Thus we may finally assume that and p(y) = _ C 2 2 1 7 ^ / 0 e s e ~ 0 n -0X1. 0 e s e ~ T 0 0 e £ , n = 1 or 3. Consequently, there are, up to conjugation, at most two irreducible representations p : 7 1 2 0 —*SU(2,C). Chapter 2. On Property I 25 It is easy to check that the preceding p(x) and p(y) satisfy (p(x)p(y))3 = 0 - 1 for n = 1 and 3, and they define two representations pn '• /120 — • SU(2,C), n = 1,3. pi and />3 e i 0 I / e s 0 are not equivalent since | | and | , 1 have different traces. • If M is a closed 3-manifold with fundamental group I\2o, then Q — Q\ n Q2 in the con-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 Q2 intersect transversally at these 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^S3^, 1//) = I120, then X(S3(K, 1//)) = 0 or ± 1 . Therefore X'(K) = 0 or ± 1 . 2.2.2 The Rohlin Invariant and the A r f Invariant Here we consider the relation of the Casson invariant with the Rohlin invariant and the Arf 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 W4. Also recall that the Z2-valued Arf invariant a of a knot K in S 3 is defined by a(K) = ]£»=i V2t-i,2i-i«2i,2t 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 3 , that is Theorem 2.2.3 ([26]) p(S3(K, 1//)) = la(K) mod 2. The Casson invariant and the Rohlin invariant are related as follows. Chapter 2. On Property 1 26 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 S3 of Arf invariant 0 has property I. Proof. By Theorem 2.2.4 and Theorem 2.2.3, \{S3(K, If I)) s p(S3(K, 1//)) = la{K) = 0 mod 2. Hence S3(K, 1//) cannot be the Poincare homology sphere by Lemma 2.1. • 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 S3 of Arf invariant 1 has property P. Note that for a knot K e 5 3 , A'(Ji') = a(K) mod 2 by Theorems 2.2.2-4. 2.2.3 The 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 3 there is an associated polynomial ^L(Z) G Z[Z]. Ambient isotopic links have identical polynomial. Axiom 2). Vv = 1 where U denotes the unknot. Axiom 3). Vs^(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) = Vi(-z). (iii) . Let V i ( z ) = ao + axz + h anzn be the Conway polynomial of a link L. Then Chapter 2. On Property I 27 di = -lk(L) if L has two components, 0 otherwise. where lk(L) denotes the linking number of L. (iv). If £ is a knot, then VL(<1 /'2-I"1 /'2) = AL(I), where Ar,(t) is the normalized Alexander polynomial of I, i.e. AL(1) = 1 and AL(I - 1 ) = AL(I)-In [41] L. Kauffman reconstructed the Jones polynomial through his bracket polynomial. The Kauffman bracket polynomial < L > (A) G Z [ A , A _ 1 ] is denned for unoriented link diagrams L, with the following defining relations. 1) . < X >= A <~> <)(>» <><>= A " 1 <x> +A <)(>, 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 2 - A" 2) < L >, 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 fL(A) = (-A)-3^ <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(* -1^4) is the Jones polynomial. Chapter 2. On Property I 28 We leave the well definedness of the Conway polynomial and the Kauffman bracket polyno-mial 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 D3. Combining with a result of Thurston's, we give a classification of cychc actions on D3 with fixed point sets of dimension 1. The Poincare sphere, first constructed by Poincare, is a very special manifold. It seems to be the first known example of a nonsimply connected closed 3-manifold with trivial first homology group. So far it is the only known homology 3-sphere with nontrivial finite fundamental group. Let D3 denote the Poincare sphere. Several descriptions of D3 can be given as follows, 1) . the manifold with the surgery presentations shown in Figure 2.3; 2) . the Seifert manifold with 3 exceptional fibers of type (5,1) (3,1) and (2,1), and cross-section obstruction —1; 3) . the Brieskorn manifold {(zu z2, z3) G C 3 ; z\ + z\ + z\ = 0, |*i| 2 + |z2|2 + N 2 = 1}; 4) . the quotient space of S3 under the free action of the binary icosahedral group, J 1 2 0 = {x,y;x2 = (xyj3 = y5, x4 = 1}. Hence the universal cover of D3 is S3 and the fundamental group of D3 is J 1 2 0 ; 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 S3 branched over the (2,3) ((2,5), (3,5)) torus knot. 7) . the boundary of the 4-manifold obtained by plumbing on the Es weighted tree. Chapter 2, On Property I 30 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 con-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 defined by p*(g) ~ g is an epimorphism, and the sequence 1 —> r —-» G 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 -1(x) and y £ p~1(g(x)). By basic covering space 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 §u 52 6 G. Then pg^ = gxpg2 = gigiP, i.e. g\h 6 G. Chapter 2. On Property I 31 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~lp and with g'(y) = x. But pgg' = gg-1p = 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'1 = ghg~lp, i.e. p ^ - 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. By 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 D3 as 3-dimensional space form, i.e. consider the orthogonal action of 50(4) on 5 3 and let D3 = 5 3 / / 1 2 o where /120 is a subgroup of 50(4) . If (-Ti2o)i> (^ 120)2 C 50(4) are subgroups isomorphic to 7i2o, it follows from [68] Theorem 4.10 and Theorem 4.11 that they are conjugate in 0(4). Consequently 5 3/(/i2o)i and 53/(/i2o)2 are isometric. Thus D3 = 53/7i2o is independent of the choice of the subgroup /120 C 50(4). T h e o r e m 2.3.1 (i). For each integer n > 1 there is an orientation preserving isometric Zn action on the Poincare 3-sphere 53//i20-(ii) . Up to conjugation by an isometry, such a Zn action is unique for each n. (iii) . If n is relative prime to 2,3 and 5, then the Zn action is free; if n is not prime to 2,3, or 5, then exactly those elements of Zn which have orders 2, 3 or 5 have fixed point sets 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 2 —» 50(4) -1+ 50(3) x 50(3) —• 1. Let ieo be a subgroup of 50(3) isomorphic to the icosahedral group. Let 7i2o = »? -1(^60 X !)• Then ii 2 o C 50(4) is isomorphic to the binary icosahedral group and acts on 5 3 fixed point freely by isometries. We shall take p : 5 3 —• D3 = 5 3 /7i 2 o as a standard universal covering of the Poincare sphere D3. (i) . We first prove the existence. Let {/") C 50(3) be a cychc group of order n. Let / € n - 1 ( l X /") and let F be the subgroup of 50(4) generated by 7i2o and / . Note that / has order n , / 7 i 2 o / - 1 = A20 and F is a group of isometries having 7i2o as a normal subgroup of index n . Let 7i2o act on 5 3 first and thus get the quotient space D3. There is an induced orientation preserving isometric cychc action on D3 of order n as follows: Let p : S3 —• D3 be the covering projection corresponding to the 7i2o action and define / : D 3 —• D3 by f(x) = pf(x) where i G D 3 and x £ p - 1 ( x ) . Then / is well defined; in fact, let x' £ p~l(x), then there is a 6 7 l 2o such that ot(x) = x ' and thus pf(x') = pf(a(x)) — p@f(x) = pf(x) where (3 = faf-1 £ I120. Similarly, using / - * , define / ' : D3 —* D3 by f'(x) = p / _ 1 ( * ) - I t i s easy to check / ' / = 1 and / / ' = 1, and thus / is an isometry of D3. As fp = pf, the order of / is n. (ii) . We now pTove the uniqueness (up to conjugation by an isometry). Let g : D3 —• D3 be an orientation preserving isometry of order n . We may assume that the geometric structure on D3 is induced from the universal covering p : 5 3 —• D3 given at the beginning of the proof. We shall prove that, up to a conjugation by an isometry of D3, the {g} action is equivalent to the {/} action given in (i). Let G ={g;g : 5 3 —• 5 3 a map with pg = gkp for some integer k}. Then G C 50(4) by our construction. By Lemma 2.3.1, 7i2o C G is a normal subgroup of index n. More explicitly, G = Ufc~i 9 A20 for some g £G with pg = gp. Claim 1. There is an element h £ 50(4) such that hGh'1 = F. Chapter 2. On Property I 33 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 50(3) . Then g" £ 1 since otherwise the kernel of n would be larger than Z2. As g' G Ieo, ( f f ' _ 1 x \){g' x g") = 1 x g" G r/(G). Suppose 5" has order m in 50(3) . Then we see n(G) = ho X {g") and thus m = n by Lemma 2.3.1. Since isomorphic subgroups of 50(3) are conjugate, there are h" G 50(3) such that h"{g"Yh"-1 = {/"}. So n(G) is conjugate to Ieo x {/"} in 50(3) x 50(3) by the element 1 X h". Let h G r / - 1 ( l x h"). Then since the kernel of n is Z 2 which is contained in both G and F, hGh.-1 = F. Note that hl^oh'1 = Iuo-C l a i m 2. There is an isometry h : D3 —• D3 such that /i{<7}/i_1 = {/}. Proof of Claim 2. Define h : D3 —• D3 by /i(x) = ph{x) where x G p~l(x). Then / i is weU defined. In fact, let x . G p _ 1 ( x ) , then there is a G /120 such that a(x) = x» and thus ph(x») — ph(a(x)) = p/3h(x) = ph(x) where (3 = / m / i - 1 G 7i2o- Similarly, using h'1, define h' : D 3 —* D3 by /i'(x) = p h - 1 ^ ) where x G p - 1 ^ ) - 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~x = h'. Now let x G £ 3 , hgh-*(x) = hgph,-1^) = hpghrl(x) = p / ^ - ^ x ) = p/ . (x) = /*p(*) = / f c (x) where / , = hgh'1 G F has order n. Hence / 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 3 I S3 9 50(3) x 50(3) x 50(3) 1 50(3) Chapter 2. On Property I 34 commutes, where the two vertical arrows denote the actions on 5 3 and 50(3) respectively and q is the quotient map defined by the standard Z2 action on S3., Note that an element g' x g" 6 50(3) X 50(3) acts on 50(3) fixed point freely iff g' is not conjugate to g" in 50(3). Also note that two elements in 50(3) with finite orders 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 J6o x {g"} acts on 50(3) freely since orders of elements in 76o can only be 2,3 and 5. Hence we have a free induced Zn action on D3. 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) with fixed point sets. This in turn induces orientation preserving Zk actions on D3 with fixed point sets. By Smith theory [11] the fixed point set of each such cychc action is a 1-sphere in D3. • Corollary 2.3.1 Let g : D3 —• D3 be an isomorphism of order n. If the fixed point set of 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 with fixed point 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 group 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-dimensional fixed point set. Now apply Theorem 2.3.1. • Chapter 2. On Property I 35 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) = G Z[t]. Since Ait-1) = I - ( P - 1 ) ( 9 - 1 ) A ( I ) , 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) = (p2 - l)(c2 - l)/24. Since p > q > 1 and p ^  3 or q ^  2, (1/2)A"(1) > (32 - 1)(22 - l)/24 = 1. Now apply Lemma 2.2.2 • Corollary 2.4.1 ([37]) Nontrivial torus knots T(p,q) have property P. Proof. As ( l / 2 ) A £ ( p i j ) ( l ) = (p2 - l)(g2 - l)/24 ^ 0 for p > q > 1, Theorem 2.2.1 (ii) applies. • Proposition 2.4.1 and Corollary 2.4.1 together give Corollary 2.4.2 Nontrivial nontrefoil torus knots have property PI. • Chapter 2. On Property I 36 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 Arf invariant is an invariant of concordance [64]. Since any slice knot is concordant with the trivial knot and the Arf 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) Arf 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 Arf invariant as well as results in section 2.2. Let Ki and K2 be knots in S3. The band-connect sum of K\ and K2 is a knot denned as follows. Separate Ki and K2 by an imbedded 2-sphere S2 C S3. Let 6 : 7 x 7 — • S3 be an imbedding such that 6_1(7i'1) = 7x0, 6_1(ijf2) = 7x1. Then join the arcs Kx - 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 K2, denoted by Ki#bK2 (Figure 2.4). 6 is called a trivial band if there exists some 2-sphere S2 imbedded in S3 such that 6(7 x 7) D S2 is a single arc, S2 fl (K\ U K2) — 9 and S2 separates Ki and Note that if 6 is trivial, then Ki#bK2 = KX#K2. Proposition 2.4.3 7/ TiTi and K2 are two knots in S3 having the same Arf invariant, then K = K\#\,K2 has property I. Chapter 2. On Property I 37 Note that if b is trivial and K\, K2 are both nontrivial then K = Ki#K2 is a satellite knot and thus has property I by Proposition 2.4.6. Note also that if b is trivial and one of two knots, say Ki, is trivial , then K = K2 has property I by Corollary 2.2.1. To prove Proposition 2.4.3 several lemmas are needed. Lemma 2.4.1 Let K = Ki#K2 be a composite knot in S3. Then A*:(0 = A*r 2 ( t ) • AK2(t)-Proof. Let F, be a Seifert surface of Ki with genus g,, i = 1,2. Then F = Fi\F2 (the boundary connect sum) is a Seifert surface of genus gx + g2 of K. The normalized Alexander polynomial of K is A K ( « ) = t-^1+3iUet(V - tVT) where V is a Seifert matrix of F. Obviously V = V i ° j where V,- is a Seifert matrix of Ki for i = 1,2. Hence AK(t) = t~toi +92)det(V-tVT) = t~^det(Vi - tVir) • t~3*det{V2 - tV2T) = AKl(t) • Ajr2(<). • For any knot K in S3, its normalized Alexander polynomial can be expressed as A;r(t ) = ao + £*.•(< - t-1)- Thus A'K(t) = £ a , ( l - i ~ 2 ) and A f c ( l ) = 0. If K = Ki#K2, then by Lemma 2.4.1, Afc(<) = A ^ ( i ) • A * a ( * ) + 2A'Kl(t) • A'K2(t) + AKl(t) • A'^t) and thus A £ ( l ) = A ^ ( l ) + A £ 2 ( l ) . Therefore we have Lemma 2.4.2 For a composite knot K = Ki#K2 in S3, X'(K) = \'{Ki) + \'{K2). • By the note given at the end of section 2.2.3, we see Lemma 2.4.3 If Ki and K2 are knots in S3 having the same (different) Arf invariant (invari-ants), then a(Ki#K2) 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 K2 be knots in S3. Then a(Ki#bK2) = a(Ki#K2). Ch&pter 2. On Property I 38 I J r Figure 2.5: T-moves Proof. In [40], L.H. Kauffman denned a Inequivalence relation for knots in S3 and showed that two knots in S3 are r-equivalent if and only if they have the same Arf invariant. The T-equivalence is denned as follows. Let A" be a knot in S 3. Take an oriented knot diagram 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 S3 is T-equivalent iff one knot can be deformed to the other 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#K2. 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 S3 having different Arf invariants, then Ki#bKi 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 S3 has been proved recently in [30]. Chapter 2. On Property I Figure 2.6: ff,#bffa « r-equivalent to Chapter 2. On Property I 40 2.4.2 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 if = 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 if be a satellite knot in S3 with if* as a nontrivial companion knot. Let N and N* be tubular neighborhoods of if and if* in S3 with N C intN*. Let E = S3 - intN, E* = S3 - intN* and E0 = N* - intN. Then E = E* U E0. Let p, X C dE and p*, X* C dE* be preferred meridian-longitude pairs of if and if* respectively. Let u> be the winding number of if in N*. Suppose that S3(K,l/l) is a manifold with fundamental group I\20- Then dN* must be compressible in S3(K,l/l) by Dehn's lemma. Let (D2,dD2) C (S3(K,l/l),dN*) be a com-pressing 2-disc. Since dN* is incompressible in E*, (D2,dD2) C (N*(K,l/l),dN*). Hence 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, if cannot be a 0-bridge braid and by Theorem 1.2.2, if 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 ifP io be the knot contained in V as shown in Figure 2.7 (a). Let if* be any knot Chapter 2. On Property J 41 p>0 p<0 x p full ^ twists ( 8 ) (b) Figure 2.7: a generalized double knot in S3 and let A 7 " be a tubular neighborhood of K" in 5 3 . Let / be an isomorphism from V to N*. Then the image K = f(Kp,o) of KPyo under / is called a generalized double knot and K* is called a companion knot of K = f(Kp>o) (Figure 2.7). Note when p = 1 this is just the usual definition of a double knot. Proposi t ion 2 A.7 Nontrefoil generalized double knots have property I. Proof. Let K be a generalized double knot in S3 and let K* be its companion knot. If 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 KPi0 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,9 is the trivial knot. C l a i m . The normahzed Alexander polynomial of KPt9 is AAy,(0 = 2pg + 1 - pq(t + I - 1). Proof of the Claim. First we calculate the Conway polynomial ^Kp,q(t) of i v P l ? by induction on the number of twists. Orient KPi9 arbitrarily. Then by Conway recursion formula, V A ' P , , _ I -Chapter 2. On Property 1 42 P,0 P,Q-1 Lp q full twists X X X - X T > ^ C K - X T q>0 q<0 ( 8 ) (b) <c) Figure 2.8: generalized twisted knot Ji ' P i 9 V ^ - P i ? = z V . ^ , where Lp is the link of two components shown in Figure 2.8 (c). The Conway polynomial of Lp with the orientation given in Figure 2.8 (c) is pz (again using Conway recursion formula inductively). Therefore we get ^KP,9 = V^,., - pz7 = VA' | ( I ,_ 2 - 2pz2 = • • • = ^Kp,o - qp*2 = 1 - pqz7- Hence the Alexander polynomial A A p ,(r) = 1 - pq{t1^2 - t~^l2)2 = 2pq -I- 1 - pq(t + i - 1 ) and the claim is proved. Simple calculation gives (l /2)A£ P (^1) = pq. Hence by Lemma 2.2.2 only when p = ±1 and q = ±1 or p = 0 or q = 0 could Kp<q have chance to ruin property I. But then Kp,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;x2 = 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. On Property I 43 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 Nontrivial nontrefoil generalized double knots have property PI. 2.4.3 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 S3 is called a periodic knot if there is an orientation preserving automorphism / of S3 with the following properties: 1) / has period n > 1, that is, fn 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 S3 by the cychc transformation group {/} generated by / induces a n-fold cychc branched covering p : S3 —• S3/{f}. Due to the positive answer to the Smith conjecture [3], the map / is a rotation of S3, S3/{f} is isomorphic to S3, the fixed point set of / is a trivial knot in S 3 and the image of the fixed point set under p is also a trivial knot in S3. 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(S3) = 5 3 . p(K) is called a factor knot of K. Lemma 2.4.5 Let K be a periodic knot in S3 with period n. If (m,nl) = 1, then S3(K,m/l) admits a Zn action with fixed point set a 1-sphere. Proof. Let N be a tubular neighborhood of the factor knot p(K) in S3 downstairs disjoint Chapter 2. On Property I 44 from the fixed point set and let E = S3 - intN. Then N = p-1(N) is a tubular neighborhood of K in 5 3 upstairs and N is invariant under the cychc action of {/}. Let E = S3 — intN. Let p, A C dE be a preferred meridian-longitude pair of p(K). Then p - 1 ( / j ) C 9/5 is a set of n copies of meridians and p - 1 ( A ) C dE is a preferred longitude of K. Let p. be one of components of p~x{p) and X = p _ 1 ( A ) . Obviously p»[/i] = [/x], p,[A] = n[A] in H\(dE). Let c be a 1-sphere in with slope m/nl. Then p _ 1 (c ) is a set of n copies of 1-spheres in dE with slope m/l. Attaching n copies of 2-disks to each element of p - 1 ( c ) and then filling the n holes with n 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 Zn action on S3(K,m/l) with fixed point set a 1-sphere. • Proposition 2.4.8 Surgery on a periodic knot K in S3 cannot give a fake Poincare sphere. Proof. By Lemma 2.4.5, S3(K,l/l) admits a Zn action with fixed point set a 1-sphere. Suppose that for some slope 1//, S3(K,l/l) has fundamental group ii2o- Then S3(K,l/l) 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 S3(K, l/l) is the honest Poincare sphere. • A n immediate consequence of Corollary 2.3.1 and Lemma 2.4.5 is Proposition 2.4.9 A periodic knot in S3 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 S3(K,l/l) is a n-fold cychc branched cover of S3(p(K), l/nl) with branch set a 1-sphere. Since niS3(p(K), l/nl) is not trivial by Chapter 2. On Property I 45 As periodic knots have property P [16], we obtain Corollary 2.4.7 Periodic knots given in Proposition 2.4-9 and 2-4-10 have property PI. • Example 2.4.1 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 J 8 is 13 - 10(i + i _ 1 ) + 5( i 2 + r a ) - ( i 3 + r 3 ) and hence get A'(8 1 8 )= 1. 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. Chapter 2. On Property I 46 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 S3 is strongly invertible if there is an orientation preserving involution of S3 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 R3 whose axis meets K in exactly two points. Proofs of the following statements can be found in [15] [5] [57]. Let K be a strongly invertible knot in S3. Then the restriction of the involution to the knot complement can be extended to an involution of the manifold S3(K,m/l) obtained by performing m//-surgery on K. For each S3(K,m/l) the quotient under this involution is the 3-sphere S3 and S3(K,m/l) is a double branched cover of S3. Moreover the branch set downstairs of this covering can be obtained by 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, S3(K,l/l) 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-Chapter 2. On Property 1 47 Proof. By Proposition 2.4.11 and Lemma 2.2.1, we only need to show that S3(K, 1) and S3(K,-1) 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 S3 is the (3,5) torus knot up to unoriented automorphisms of S3 and thus is either the right hand or the left hand (3,5) torus knot. The branched sets corresponding to S3(K, 1) and S3(K, -1) , denoted by K\ and K-\, can 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 Kx, K-\, U and LQ have diagrams differing only at the site 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_, — zVi0 = 0. Since K\ and K.\ are right hand or left 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 un-oriented Ki, K-i, U and LQ, we have { < Ki > = A < Lo > +A-1 < U >, < K-i >= A ' 1 < LQ > +A < 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 fLo{A) = ( - A ) " 3 n < LQ >, fv{A) = Ch&pter 2. On Property I 48 (-A)~3n < U >= 1, fKl(A) = ( - A ) " 3 ^ 1 ) < Ki > and / « • _ , ( A ) = ( - A ) ~ 3 ( N - 1 ) < K-i >. Substituting them into (*) above, we have J -A*fKl(A) = fLo(A)-rA->, (**)< -A-'fK_1(A) = fL0(A) + A2 Eliminating / L 0 , we get A 2 / K J ( A ) - A - 2 / * - ^ - ^ ) = A 2 - A - 2 . Hence we have either (i) . / R - J = = 1 if i f i is ambient isotopic to i f _ i ; or (ii) . A2fxl{A) - A^fKiiA'1) = A2 - A'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 3 , 5 3(if, m/l) = S3(K, —m/l). Hence we obtain Corollary 2.4.8 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 in S3 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. On Property I 49 FiguTe 2.10: a pretzel knot of type K(pi,- •••,pm) A pretzel knot of type (j>\,P2, • • ••,Pm) in S3 is a knot having a knot diagram as shown in Figure 2.10 where each box BPi denotes a two-strand braid with pi half-twists. First we show Proposi t ion 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 3 , K C S and S- K is connected. Let A, B be the closure of the two complements of S in S3. Then both A and B are standard handbodies of genus two. By homological arguments it can be 6hown that E\(A, S-K) = Z 2 , Ei(B, 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 3 and let E = S3 - tni(JV). Let /z,A C dE be a meridian and a preferred longitude of K. Then E\(dE) = Z[p] + Z[X] and [A] = 0 in Chapter 2. On Property I 50 Let X be a boundary component of an annular neighborhood of K in S. Then [X] = ± 2 ( P + «)[M] in ffiPO = and [X] = ± 2 ( p + g)[/x] + [A] in Hx(dX) = Z[/*] + Z[A] ([71] [72]). We are now going to show that S3(K, 1) has infinite fundamental group (the case of —1 surgery can be proved in exactly the same way). It follows from Van Kampen's theorem that ir\S3(K,l) is isomorphic to the free product of *i(A) and iti(B) amalgamated along ITI{S - K) with additional relation X ± 1 ( a & ) ± 2 ( p + » ) : k l = 1, where a 6 *\(A\ b 6 JTI(B). By first annihilating iri(S—K) and then abelianizing it\(A) and it\(B), we obtain a homomorphism from TnS3(K, 1)onto H^A,S-K)*HiiBiS-K)/ < (ab)±2^)±1 >= Z2*Zd/ < (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 Z2 * Zdj < (o6) ± 2 ( p + ' ) ± 1 > is not a finite group. Therefore ic\S3(K, 1) cannot be finite. • 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, Proposi t ion 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(2,2m + 1) torus knot as a factor knot (Figure 2.11). Now apply Proposition 2.4.10. • Note that W . Ortmeyer showed in [62] that R3 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 up to 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 2 61 6 2 63 7i 7 2 7 3 7 4 7 5 7 6 7 7 A' 1 -1 3 - 2 - 2 1 1 6 3 5 4 4 1 1 knot 8i 8 2 8 3 8 4 85 8 6 87 8 8 89 810 811 812 8l3 814 A' - 3 0 -4 - 3 - 1 - 2 2 2 - 2 3 - 1 - 3 1 0 knot 8» 8l6 817 8is 819 820 821 9i 9 2 9 3 9 4 9 5 9 6 9 7 A' 4 1 1 1 5 2 0 10 4 9 7 6 7 5 knot 9 8 9 9 9io 9 n 9l2 9l3 9 u 9l5 9l6 9l7 9l8 9l9 9 2 0 921 A' 0 8 8 4 1 7 -1 2 6 - 2 6 - 2 2 3 knot 9 2 2 9 2 3 924 9 2s 9 2 6 9 2 7 9 2 8 9 2 9 93o 931 9 3 2 9 3 3 9 3 4 9 3 5 A' - 1 5 1 0 0 0 -1 -1 -1 2 1 1 -1 7 knot 9 3 6 9 3 7 9 3 8 9 3 9 9 4 0 9 « 9 4 2 9 4 3 9 4 4 9 4 5 9 4 6 9 4 7 9 4 8 9 4 9 A' 3 - 3 6 2 -1 0 - 2 -1 0 2 - 2 -1 3 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, 63, 8s, 8 i 8 has been shown in section 2.4.2, 2.4.4, 2.4.5, 2.4.3 respectively. Except for the knots 817, 9 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. Chapter 2. On Property I 52 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 C o n c l u d i n g Remarks and O p e n Problems Let i f be a knot in 5 3 and let E = S 3 — intN(K) be its knot complement. Let F be a closed connected incompressible surface in E. Note that F is necessarily oritentable and it separates E into two components, say E\ and E2, that is, E = E\ U E2, E\ fl E2 = dE\ fl dE2 = F. Assume that E2 is the component which contains dE. The surface F is called a m-surface if there is an annulus A properly embedded in E2 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 A2 properly embedded in E2 with dA\ — sx U mx and dA2 = s2\J m2 such that si and s2 are nonisotopic simple closed curves in F and that mj 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 S3(K, m/l) obtained by a nontrivial surgery on K. Hence knots with 2m-surfaces have property PI by Dehn's lemma. Quest ion 2.5.1. Let K C S3 be a knot with a m-surface F. Is it true that F remains incompressible in each manifold S3(K,m/l) with m/l ^ 1/0? Recall that a Montesinos knot of type (pi/qi,...,pn/<7n) is a knot having a knot diagram as shown in Figure 2.12 where each Tpi/qi denotes the rational tangle of type pi/qt. In [60] U . Oertel showed that a Montesinos knot of type ( p i / g i , . . -,pn/<ln) with n > 4, qi > 3, i = 1 , . . . , n, is a knot with 2m-surface. Therefore this family of Montesinos knots have property PI. Question 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? Chapter 2. On Property I 53 Figure 2.12: a Montesinos knot of type ( p i / g \ , p n / 9 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. Question 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 S3. 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. Recently S. Bleiler and C. Hodgson have shown [6] that if a hyperbolic knot in S3 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 2 / J 2 is |mi/ 2 - rn 2/i|). Hence if 1// surgery 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 3 , i l might be helpful to consider the approach suggested by the following two questions. Question 2.5.4. If S3(K, 1 //) is a fake Poincare sphere, is it homotopy equivalent to the honest 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, in S3 downstairs of the covering p : S3 —•* S3/{f) becomes a torus knot of type ( ± 3 , 5 ) or ( ± 2 , 5 ) or ( ± 2 , 3 ) respectively after performing 1 or —1 surgery on p(K) in S3. From the discussion in section 2.4.4, we see that to solve property I for strongly invertible knots in 5 3 , it is enough to solve the following 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 S3 such that some nontrivial surgery on K gives a manifold with finite fundamental group? C h a p t e r 3 O n B o u n d a r y Slopes 3.1 Introduction Let K in S3 be a nontrivial knot, let N(K) be a tubular neighborhood of K in S3, and let E — S3 — 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]. Question. ([33]) Is it true that <p(K) C Z U {1/0} for every knot K in S3 ? 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 S3 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 of 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 irxS3(K,m) 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) such that (1). || • || 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 , ± / ) € Hx{dE,Z). (2) . Define n = min{\\(m,l)\\;(m,l) G Hi(dE,Z)-(0,0)} and consider the ball J5 of radius n in Hi(dE,R). Then i? is a compact, convex, finite sided polygon which is symmetric about the origin (i.e. -B = B). Note that intB fl Hx(dE,Z) = (0,0). (3) . For any vertex of B, there is a primitive element (m,/) G Hx(dE, Z) such that (m,/) 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 S3(K,m/l) 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 Hx(dE, Z) 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 in the way described in (3). By 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 vx — (si,s2) and v2 = ( i i , t 2 ) be the two vertices of T. Let L be the line in H\(dE,R) passing through points {(m, 1); m G Z}. Case 1. T is not parallel to L. Then one of the vertices of T, say vx = (si,s2) must lie above the line L in the sense that s2 > 1. Such a vertex certainly determines a nonintegral boundary slope in the way described in (3). Case 2. T is parallel to L. Then m G T (as an interior point) and vx = ( « i , l ) , v2 = ( r j , l ) . We may assume that si < m < m + 1 < tx. We must have m — l < s i < * i < m + 2 since otherwise the area of Chapter 3. On Boundary Slopes 58 B would be large than 4, violating (5). Now both v\ and v2 determine nonintegral boundary slopes as required. • 3.3 P r o o f of 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 . Fintushel and 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 S3 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 S3 and hence S3 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 S3(K,m/l) and the quotient under the corresponding involution is S3. Moreover the branched set in S3 of the corresponding double covering can be obtained by 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 S3 corresponding to 18 and 19 surgeries 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. On boundary Slopes 60 isocopy 21 preferred l a c c i t u d e 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 18-surgery 19/8 2-bridge knot (b) branch set corresponding to 19 -surgery 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(18,5) and 1(19,8) . • 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. On Boundary Slopes 62 3.4 Properties o f <p(K) and O p e n Problems 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 S3. 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}. Question 3.4.1. Is it true that for a nontorus knot K in 5 3 , Iv -^SQI > 2? T h e o r e m 3.4.2 ([32]) <p(K) is a finite set for any knot in S3. 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 S3) and pq (the slope of the cabling annulus) can be 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 K2n 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 K2n 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 S3 and let L C H\{dM, R) be the semi-line which starts from (0,0) and passes (m, /). Question 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 ' if admits two nontrivial cyclic surgeries' is reduced to ' if admits one nontrivial cyclic surgery'? If the answer is yes, then all if2n (M > 1) have nonintegral boundary slopes. Example 1.4.2 shows that each Berge-Gabai knots Jn admits two nontrivial integral surg-eries. Question 3.4.4. Is Jn not sufficiently large? 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