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Topics on Dehn Surgery TOPICS ON DEHN SURGERY By Xingru Zhang B.Sc. of Mathematics, Nanjing Institute of Posts and Telecommunications A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES MATHEMATICS We accept this thesis as conforming to the required standard THE 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 in 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. ii Table of Contents Abstract " List of Figures v Acknowledgements V1 Introduction v" 1 On Cyclic Surgery 1 1.1 Introduction 1 1.2 Preliminaries 5 1.2.1 CM. 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, „.,pn/on) 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. Final thanks go to my family, especially to my wife, Lijuan Zhang, for their emotional support. vi Introduction One of the basic methods to construct closed orientable 3-manifolds is by Dehn surgery on knots or links in the 3-sphere S3, which was introduced by M. Dehn in 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] in 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 in 53, 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 if the complement of a hyperbolic knot contains no incompressible nonboundary parallel closed surfaces, then again except for finitely many cases Dehn surgeries on the knot yield hyperbolic manifolds that do not contain incompressible closed surfaces. For a satellite knot, nonboundary parallel incompressible tori in the knot complement will remain incompressible in 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 53 can yield the Poincare homology 3- sphere? More generally which Dehn surgery on which knot in S3 can yield a manifold with fundamental group I\2Q, the binary icosahedral group? Topic 3. Axe there nonintegral boundary slopes for knots in 53? 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 in 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 in 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. viii Chapter 1 On Cyclic Surgery 1.1 Introduction We work in all three chapters in the PL category. A PL 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.
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