Integral Finite Surgeries on Knots in S³ Citation Gu, Liling (2014) Integral Finite Surgeries on Knots in S³. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/E7VQ-SV60. https://resolver.caltech.edu/CaltechTHESIS:06012014-162932302 Abstract Using the correction terms in Heegaard Floer homology, we prove that if a knot in S 3 admits a positive integral T-, O-, or I-type surgery, it must have the same knot Floer homology as one of the knots given in our complete list, and the resulting manifold is orientation-preservingly homeomorphic to the p-surgery on the corresponding knot. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Heegaard Floer Homology, Low-dimensional Topology Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Ni, Yi Thesis Committee: Ni, Yi (chair) Graber, Thomas B. Liu, Yi Marcolli, Matilde Defense Date: 29 May 2014 Record Number: CaltechTHESIS:06012014-162932302 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06012014-162932302 DOI: 10.7907/E7VQ-SV60 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8460 Collection: CaltechTHESIS Deposited By: Liling Gu Deposited On: 03 Jun 2014 18:54 Last Modified: 05 Jul 2022 19:07 Thesis Files Preview PDF - Final Version See Usage Policy. 519kB Repository Staff Only: item control page

Integral finite surgeries on knots in S 3 Thesis by Liling Gu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2014 (Defended May 29, 1014) ii c 2014 Liling Gu All Rights Reserved iii Acknowledgements First and foremost, let me thank my advisor, Yi Ni, a kind and patient mentor, for his countably infinite ideas and intuitive suggestions, on this topic and on topology in general. I owe also a great debt of gratitude to Danny Calegari. Danny has taught many Caltech graduate students, me included, most of what we know about algebraic topology and hyperbolic geometry. I would like to thank the physics department for kindly letting me transfer to the mathematics department, and the mathematics department for accepting me as a graduate student. I would like to thank my friends and fellow graduate students in mathematics, with whom I spent many late nights doing problem sets and studying for qualifying exams, among them Nakul Dawra, Foo Yee Yeo, Dapeng Zhang, Brian Skinner, Daiqi Linghu, and Branimir Ćaćić. Thanks to Nakul Dawra in particular, for having been a wonderful officemate, friend, and colleague. I would like to thank John Berge for kindly sending me his unpublished papers, and Xingru Zhang for pointing out that I may use Dean’s twisted torus knots. Finally, I would like to thank my roommates Liuyan Zhao and Yu Zhao, who helped me to live a diverse and balanced life. iv To my parents v Abstract Using the correction terms in Heegaard Floer homology, we prove that if a knot in S 3 admits a positive integral T-, O-, or I-type surgery, it must have the same knot Floer homology as one of the knots given in our complete list, and the resulting manifold is orientation-preservingly homeomorphic to the p-surgery on the corresponding knot. vi Contents Acknowledgements iii Abstract v 1 Introduction 1 2 Preliminaries on 3-Manifolds 3 2.1 Dehn Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Seifert-Fibered Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Heegaard Floer Homology 3.1 5 Heegaard Floer Homology . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Heegaard Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.2 Spinc Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.3 Floer Chain Complexes . . . . . . . . . . . . . . . . . . . . . . 6 3.1.4 Absolute Q-grading . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Knot Floer Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Correction Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Proof Strategy 12 5 Obstructions When p Is Large 15 6 Constructions When p Is Small 23 6.1 Dean’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Montesinos Construction . . . . . . . . . . . . . . . . . . . . . . . . . 27 6.3 Berge’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 vii 7 Results 29 Bibliography 40 viii List of Figures 6.1 Link for K(107, 30; 5) . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7.1 Pretzel knot P (−2, 3, 7) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 Pretzel knot P (−2, 3, 9) . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.3 Mirror image of K(1, 1, 0) from section 4 of [8] . . . . . . . . . . . . . . 32 7.4 Twisted torus knot K(11, 3, 2, 1) from [6] . . . . . . . . . . . . . . . . . 32 7.5 Berge knot K(39, 16; 16) . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.6 Berge knot K(45, 19; 8) . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7.7 K3∗ from [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.8 K2] from [4] 34 7.9 Mirror image of the P/SFd KIST IV knot with (n, p, , J1 , J2 ) = (−2, 1, 1, −4, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . from [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.10 P/SFd KIST III knot with (h, k, h0 , k 0 , J) = (−5, −3, −2, −1, 1) from [2] 35 7.11 Mirror image of twisted torus knot K(2, 3, 5, −3) from [12] . . . . . . . 35 7.12 Berge knot K(71, 27; 11) . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.13 Twisted torus knot K(2, 3, 5, 3) from [12] . . . . . . . . . . . . . . . . . 36 7.14 P/SFd KIST V knot with (n, p, , J1 , J2 ) = (1, −2, −1, 2, 2) from [2] . . 36 7.15 Mirror image of twisted torus knot K(3, 4, 7, −2) from [12] . . . . . . . 36 7.16 Mirror image of twisted torus knot K(2, 3, 5, −4) from [12] . . . . . . . 37 7.17 Twisted torus knot K(2, 3, 5, 4) from [12] . . . . . . . . . . . . . . . . . 37 7.18 Twisted torus knot K(3, 4, 7, 2) from [12] . . . . . . . . . . . . . . . . . 37 7.19 Mirror image of twisted torus knot K(3, 5, 8, −2) from [12] . . . . . . . 38 7.20 Mirror image of the P/SFd KIST V knot with (n, p, , J1 , J2 ) = (−3, −2, −1, 2, 2) 7.21 from [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Mirror image of twisted torus knot K(2, 3, 5, −5) from [12] . . . . . . . 38 ix 7.22 Twisted torus knot K(2, 3, 5, 5) from [12] . . . . . . . . . . . . . . . . . 38 7.23 Mirror image of twisted torus knot K(2, 5, 7, −3) from [12] . . . . . . . 39 7.24 Twisted torus knot K(3, 5, 8, 2) from [12] . . . . . . . . . . . . . . . . . 39 7.25 Twisted torus knot K(2, 5, 7, 3) from [12] . . . . . . . . . . . . . . . . . 39 x List of Tables 6.1 Candidates of Berge knots with T-, O-, and I-type surgeries . . . . . . 24 6.2 Berge knots with T-, O-, and I-type surgeries . . . . . . . . . . . . . . 26 7.1 Torus knots and satellite knots with T-, O-, and I-type surgeries . . . 29 7.2 Hyperbolic knots with T-, O-, and I-type surgeries . . . . . . . . . . . 30 1 Chapter 1 Introduction In the early 1960s, Wallace [28] and Lickorish [11] proved independently that any closed, orientable, connected 3-manifold can be obtained by performing Dehn surgery on a framed link in the 3-sphere. One natural question is which manifolds can be obtained by some surgery on a knot. In this paper we consider the manifolds with finite noncyclic fundamental groups. By Perelman’s resolution of the Geometrization Conjecture [22, 23, 24], the manifolds with finite fundamental group are spherical space forms. They fall into five classes: lens spaces; prism manifolds; T-, O-, and I-type spherical space forms. Berge [1] constructed a list of knots which yield lens space surgeries, and he conjectured that it is complete. Greene [9] proved that if a p-surgery along a knot produces a lens space L(p, q), there exists a p-surgery along a Berge knot with the same knot Floer homology groups as L(p, q). In this paper we focus on knots with finite noncyclic surgeries. According to Thurston [27], a knot is either a torus knot, a hyperbolic knot, or a satellite knot. Moser [14] classified all finite surgeries on torus knots, and Bleiler and Hodgson [3] classified all finite surgeries on cables of torus knots. Boyer and Zhang [4] showed that a satellite knot with a finite noncyclic surgery must be a cable of some torus knot. There is also some progress about hyperbolic knots. Doig [7] proved that there are only finitely many spherical space forms which come from a p/q-surgery on S 3 for a fixed integer p. From now on we will only consider Dehn surgeries on hyperbolic knots. Suppose M is a 3-manifold with torus boundary, α is a slope on ∂M . Let M (α) be the Dehn filling along α. If M is hyperbolic, Thurston’s Hyperbolic Dehn Surgery 2 Theorem says that at most finitely many of the fillings are nonhyperbolic. These surgeries are called exceptional surgeries. In [4], Boyer and Zhang showed that if M is hyperbolic, M (α) has a finite fundamental group and M (β) has a cyclic fundamental group, then |∆(α, β)| ≤ 2. In particular, if the p/q-surgery on a hyperbolic knot 3 K ⊂ S 3 , denoted SK (p/q), has a finite fundamental group, then |q| ≤ 2. For |q| = 2, Li and Ni [10] proved that K has the same knot Floer homology as either T (5, 2) or a cable of a torus knot (which must be T (3, 2) or T (5, 2)). From now on we will only consider integral surgeries on hyperbolic knots. Taking the mirror image of a knot K if necessary, we may assume p > 0. We consider here all T-, O-, and I-type spherical space forms. In this paper, all manifolds are oriented. If Y is an oriented manifold, then −Y denotes the same manifold with the opposite orientation. Let T be the exterior of the right-hand trefoil, then T(p/q) is the manifold obtained by p/q-surgery on the right-hand trefoil. It is well-known that any T-, O-, or I-type manifold is homeomorphic to some ±T(p/q) (see Lemma 4.0.4). Our main result is the following theorem: Theorem 1.0.1. Suppose that K is a knot in S 3 , and that the p-surgery on K is a T-, O-, or I-type spherical space form for some integer p > 0, then K has the e in table 7.1 or table 7.2, and the same knot Floer homology as one of the knots K resulting manifold is orientation-preservingly homeomorphic to the p-surgery on the e corresponding knot K. Chapter 2 contains some preliminaries on 3-manifolds. Chapter 3 contains a brief summary of Heegaard Floer Homology, Knot Floer Homology and correction terms. After setting up the background, Chapter 4 briefly explains the proof strategy of Theorem 1.0.1 which is carried out in the following two chapters. Chapter 5 proves that there is no surgeries on knots which yield T-, O-, or I-type spherical space forms when the surgery coefficient is large enough, and gives a list of candidate coefficients. For each candidate coefficient, Chapter 6 then gives a knot which yields a spherical space form. Finally, Chapter 7 summarizes all the results and draws all the hyperbolic knots with T-, O-, or I-type surgeries. 3 Chapter 2 Preliminaries on 3-Manifolds 2.1 Dehn Surgery Dehn surgery is probably the most common way of constructing 3-manifolds. Let L = L1 ∪ ... ∪ Ln be an oriented link embedded in an oriented 3-manifold Y . A Dehn surgery removes disjoint open neighborhoods Ni of Li and glues back the disjoint union of n solid tori, which gives a 3-manifold (M − (N1 ∪ ... ∪ Nn )) ∪φ (∪S 1 × D2 ), n where φ is a union of homeomorphisms, each of which take a meridian curve of ∂(S 1 × D2 ) onto a specified curve Ji in ∂Ni . By Wallace [28] and Lickorish [11], each closed, orientable, connected 3-manifold can be obtained by surgery on a link in S 3 . We will only consider Y = S 3 here. For each link component Li , choose a meridian µi and a longitude λi with linking number 1. Every simple closed curve Ji on ∂Ni is isotopic to pi µi + qi λi , and the surgeried manifold is denoted by SL3 (p1 /q1 , ..., pn /qn ). 2.2 Seifert-Fibered Space A Seifert-fibered space is a 3-manifold fibered by circles, where any fiber has a neighborhood which is a disjoint union of circles that forms a standard fibered torus. A standard fibered torus corresponding to a pair of coprime integers (a, b) with a > 0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2πb/a. If a = 1 the middle fiber is called regular, while if a > 1 the middle fiber is called exceptional, and a is called the multiplicity of the fiber. A compact Seifert-fibered space has finitely many exceptional fibers with multiplicities a1 , ..., ap . 4 The set of fibers forms a 2-dimensional orbifold, and is called base orbifold of the Seifert-fibered space. Denote a 2-orbifold which is topologically X with p cone points of indices a1 , ..., ap by X(a1 , ..., ap ). By Perelman’s resolution of the Geometrization Conjecture [22, 23, 24], the manifolds with finite fundamental group are spherical space forms. They fall into five classes, those with cyclic π1 and those with finite π1 based on the four isometries of a sphere: Theorem 2.2.1. (Seifert [26]). If Y 3 is closed, oriented and Seifert-fibered with finite but noncyclic fundamental group, then it is one of: 1. a D-type spherical space form, or a prism manifold with base orbifold S 2 (2, 2, n), 2. a T-type spherical space form, or a tetrahedral manifold with base orbifold S 2 (2, 3, 3), 3. a O-type spherical space form, or a octahedral manifold with base orbifold S 2 (2, 3, 4), 4. a I-type spherical space form, or a icosahedral manifold with base orbifold S 2 (2, 3, 5). 5 Chapter 3 Heegaard Floer Homology 3.1 Heegaard Floer Homology Heegaard Floer homology was introduced by Ozsváth and Szabó [17]. Given a closed oriented 3-manifold Y and a Spinc structure s ∈ Spinc (Y ), one can define the Heegaard d (Y, s), HF + (Y, s),..., which are invariants of (Y, s). Floer homology HF 3.1.1 Heegaard Diagrams For a closed oriented three-manifold Y , we can associate a pointed Heegaard diagram (Σg , α1 , ..., αg , β1 , ..., βg , z) to it, and Y = U0 ∪Σg U1 . Σg is an oriented closed surface of genus g and is the boundary of handlebodies U0 and U1 , α = (α1 , ..., αg ), β = (β1 , ..., β2 ) are attaching curves for U0 and U1 , and z ∈ Σg −α−β. Define Symg (Σg ) = Σg × · · · × Σg /Sg , where Sg is the symmetric group on g letters. The attaching curves induce two g-dimensional tori in Symg (Σg ), Tα = α1 × · · · × αg and Tβ = β1 × · · · × βg . The two tori intersect transversally in finitely many points in Symg (Σg ). For a pair of intersection points x, y ∈ Tα ∩Tβ , let π2 (x, y) denote the set of homotopy classes of Whitney disks connecting x and y, where a Whitney disk is a continous map u : D ⊂ C → Symg (Σg ) with u(−i) = x, u(i) = y, u(∂D|Re(z)≥0 ) ⊂ Tα , u(∂D|Re(z)≤0 ) ⊂ Tβ . For any point w ∈ Σg −α−β, let nw : π2 (x, y) → Z denote the algebraic intersection number nw (φ) = #φ−1 ({w} × Symg−1 (Σg )). A complex structure on Σ induces a complex structure on Symg (Σg ). For a given 6 homotopy class φ ∈ π2 (x, y), let M(φ) denote the moduli space of holomorphic representatives of φ. There is an R action on M(φ) that corresponds to the group c of complex automorphisms of the unit disk that preserves i and −i. Define M(φ) = M(φ)/R. The moduli space M(φ) has an expected dimension called the Maslov index µ(φ). 3.1.2 Spinc Structures Define Spinc structures over Y , denoted by Spinc (Y ), as nowhere vanishing vector fields modulo equivalent relation where two vector fields are homologous if they are homotopic outside a ball. Let f be a Morse function on Y compatible with the attching curves. Then each x ∈ Tα ∩ Tβ determines a g-tuple of trajectories for ∇f connetcting index one critical points to index two critical points. Similarly z gives a trajectory connecting the index zero critical point to the index three critical point. Deleting tubular neighborhoods of these g + 1 trajectories from Y , ∇f defines a nonvanishing vector field. Since each trajectory connects critical points of different parities, the vector field can be extended over Y . Denote the homology class of the vector field constructed this way by sz (x). This defines a map sz : Tα ∩Tβ → Spinc (Y ). There is a one-to-one correspondence between Spinc (Y ) and H 2 (Y, Z). 3.1.3 Floer Chain Complexes d (α, β, s) be the free Abelian group Choose a Spinc structure s ∈ Spinc (Y ). Let CF generated by the points x ∈ Tα ∩ Tβ with sz (x) = s. The group can be endowed with a relative grading gr(x, y) = µ(φ) − 2nz (φ), where φ ∈ π2 (x, y). The differential is definied by ∂x = X X c #(M(φ)) ·y y∈Tα ∩Tβ φ∈π2 (x,y),µ(φ)=1,nz (φ)=0 c By examing the Gromov compactification of M(φ) for nz (φ) = 0 and µ(φ) = 2, d (α, β, s), ∂) is a chain complex. Denote the it is proved in [17] that ∂ 2 = 0. So (CF d (α, β, s). Moreover, it is proved in [17] homology groups of this chain complex by HF that the homology groups are isomorphic for different pointed Heegaard diagrams of d (Y, s) = HF d (α, β, s). Y , so it is an invariant of (Y, s) and is denoted by HF 7 Similarly, we can define HF ∞ (Y, s). Let CF ∞ (α, β, s) be the free Abelian group generated by pairs [x, i] where x ∈ Tα ∩ Tβ , i is an integer, and sz (x) = s. The group can be endowed with a relative grading gr([x, i], [y, j]) = µ(φ) − 2nz (φ) + 2i − 2j, where φ ∈ π2 (x, y). The differential is definied by ∂[x, i] = X X c #(M(φ)) · [y, i − nz (φ)] y∈Tα ∩Tβ φ∈π2 (x,y),µ(φ)=1 There is an isomorphism U on CF ∞ (α, β, s) given by U [x, i] = [x, i − 1]. Let CF − (α, β, s) be the subgroup of CF ∞ (α, β, s) which is generated by [x, i] where i < 0, and let CF + (α, β, s) be the quotient group. The short eaxat sequence of chain complexes ı π 0 −−−→ CF − (α, β, s) −−−→ CF ∞ (α, β, s) −−−→ CF − (α, β, s) −−−→ 0 induces a long exact sequence ı π · · · −−−→ HF − (Y, s) −−−→ HF ∞ (Y, s) −−−→ HF + (Y, s) −−−→ · · · . Similarly, ı̂ U d (α, β, s) −−− 0 −−−→ CF → CF + (α, β, s) −−−→ CF + (α, β, s) −−−→ 0 induces ı π d (Y, s) −−−→ HF + (Y, s) −−−→ HF + (Y, s) −−−→ · · · . · · · −−−→ HF Moreover, a cobordism between two 3-manifolds induces functorial maps between long exact sequences of Heegaard Floer homologies of the two 3-manifolds. More precisely, let HF ◦ (Y, s) denote one type of Heegaard Floer homology, and denote the ◦ map induced by cobordism W from Y1 to Y2 by FW,s : HF ◦ (Y1 , s1 ) → HF ◦ (Y2 , s1 ), where si = s|Yi for i = 1, 2. We have the following commutative diagrams: ı π · · · −−−→ HF − (Y1 , s1 ) −−−→ HF ∞ (Y1 , s1 ) −−−→ HF + (Y1 , s1 ) −−−→ · · ·    ∞  −  +  F FW,s F W,s y y W,s y ı π · · · −−−→ HF − (Y2 , s2 ) −−−→ HF ∞ (Y2 , s2 ) −−−→ HF + (Y2 , s2 ) −−−→ · · · and ı π d (Y1 , s1 ) −−− · · · −−−→ HF → HF + (Y1 , s1 ) −−−→ HF + (Y1 , s1 ) −−−→ · · ·     +  +  FW,s FW,s FbW,s y y y ı π d (Y2 , s2 ) −−− · · · −−−→ HF → HF + (Y2 , s2 ) −−−→ HF + (Y2 , s2 ) −−−→ · · · 8 3.1.4 Absolute Q-grading When s is torsion, there is an absolute Q-grading which lifts the relative Z-grading d (Y, s). It is uniquely characterized by the following properties: on HF 1. ı̂, ı and π preserve the absolute grading, d (S 3 , 0) is supported in absolute grading zero, 2. HF 3. if W is a cobordism from Y1 to Y2 , and ξ ∈ HF ∞ (Y1 , s1 ), then ∞ gr(FW,s (ξ)) − gr(ξ) = c1 (s)2 − 2χ(W ) − 3σ(W ) 4 where si = s|Yi for i = 1, 2. 3.2 Knot Floer Homology In this section, we restrict ourselves to knots in S 3 . Given a Heegaard diagram (Σg , α1 , ..., αg , β1 , ..., βg ) for S 3 with two basepoints w and z, this data specifies a knot in S 3 . Connect w and z by a unique curve a ( up to homotopy ) in Σg − α and also by a curve b in Σg − β. Pushing a and b into U0 and U1 respectively, we obtain a knot K in S 3 . Conversely, every knot can be represented by a two-pointed Heegaard diagram. For a knot K ⊂ S 3 , we can thicken a projection of K ∈ S 2 (= R2 + ∞) to get a handlebody U0 , and take U1 be the complement of U0 . Mark a point x on the boundary of the unbounded region, U0 ∩ S 2 minus the outermost curve are α attaching curves. Each double point gives rise to a β curve, these β curves together with a meridian µ near x are β attaching curves. Choose two points on the two sides of µ to be w and z. This gives rise to a two-pointed Heegaard diagram representing K. Let K be a knot in S 3 and (Σg , α1 , ..., αg , β1 , ..., βg , w, z) be a compatible twopointed Heegaard diagram. Let C(K) be the free Abelian group generated by the points x ∈ Tα ∩ Tβ . The differential is definied by ∂K x = X X y∈Tα ∩Tβ φ∈π2 (x,y),µ(φ)=1,nz (φ)=nw (φ)=0 c #(M(φ)) ·y 9 b \ (C(K), ∂K ) is a chain complex. Its homology HF K(K) is independent of the two-pointed Heegaard diagram representing K. The group can be endowed with two absolute gradings: Maslov grading and Alexander grading. The Maslov grading is the absolute Q-grading coming from the Maslov grading induced by w, since (Σg , α1 , ..., αg , β1 , ..., βg , w) is S 3 . The Alexander grading is the unique function satisfying A(x) − A(y) = nz (φ) − nw (φ) where φ ∈ π2 (x, y) and #{x ∈ Tα ∩ Tβ |A(x) = i} ≡ #{x ∈ Tα ∩ Tβ |A(x) = −i} (mod 2). We write \ \ HF K(K) = ⊕i,j HF K i (K, j), where i is Maslov grading and j is Alexander grading, induced from the Chain complexes. 3.3 Correction Terms When Y is a rational homology sphere, Ozsváth and Szabó [16] defined a correction term d(Y, s) ∈ Q, which is the minimal grading of any non-torsion element in the image of HF ∞ (Y, s) in HF + (Y, s). The correction terms have the following symmetries: d(Y, s) = d(Y, Js), d(−Y, s) = −d(Y, s), (3.1) where J : Spinc (Y ) → Spinc (Y ) is the conjugation. Suppose that Y is an integral homology sphere, K ⊂ Y is a knot. Let YK (p/q) be the manifold obtained by p/q-surgery on K. Ozsváth and Szabó defined a natural identification σ : Z/pZ → Spinc (YK (p/q)) [16, 21]. For simplicity, we often use an integer i to denote the Spinc structure σ([i]), when [i] ∈ Z/pZ is the congruence class of i modulo p. d (Y ) = |H1 (Y )|. Examples A rational homology sphere Y is an L-space if rankHF of L-spaces include spherical space forms. For a knot K ⊂ S 3 that admits an intergral \ L-space surgery, HF K is determined explicitly from the Alexander polynomial of K by the following theorem. 10 Theorem 3.3.1. (Ozsváth and Szabó [20]) If a knot K ⊂ S 3 admits an intergral L-space surgery, then there is an increasing sequence of Alexander gradings n−k < \ · · · < nk with the property that ni = −n−i , HF K(K, j) = 0 unless j = ni for some \ i, in which case HF K(K, j) ∼ = Z and it is supported entirely in Maslov grading δi , where      0 δi = δi+1 − 2(ni+1 − ni ) + 1     δi+1 − 1 if i = k if k − i is odd if k − i > 0 is even The information about the Heegaard Floer homology of an L-space is completely encoded in its correction terms. Let L(p, q) be the lens space obtained by p/q-surgery on the unknot. The correction terms for lens spaces can be computed inductively as follows: d(S 3 , 0) = 0, d(−L(p, q), i) = 1 (2i + 1 − p − q)2 − − d(−L(q, r), j) 4 4pq (3.2) where 0 ≤ i < p + q, r and j are the reductions of p and i modulo q, respectively. For example, using the recursive formula (3.2), we can get d(L(3, q), i) =   (1, −1, −1) 2 6 6 q = 1, i = 0, 1, 2  (1, 1, −1) q = 2, i = 0, 1, 2 6 6 2   ( 3 , 0, − 1 , 0) q = 1, i = 0, 1, 2, 3 4 4 d(L(4, q), i) =  (0, 1 , 0, − 3 ) q = 3, i = 0, 1, 2, 3 4 4   q = 1, i = 0, 1, 2, 3, 4 (1, 15 , − 15 , − 51 , 15 )      ( 2 , 2 , − 2 , 0, − 2 ) q = 2, i = 0, 1, 2, 3, 4 5 5 5 5 d(L(5, q), i) =   ( 25 , 0, 52 , − 52 , − 25 ) q = 3, i = 0, 1, 2, 3, 4     q = 4, i = 0, 1, 2, 3, 4. (− 51 , 15 , 15 , − 15 , −1) (3.3) (3.4) (3.5) From [21], if we know the knot Floer homology, then we can compute the Heegaard Floer homology of all the surgeries on K. In particular, if the p/q-surgery on K ⊂ S 3 3 is an L-space surgery, where p, q > 0, then the correction terms of SK (p/q) can be 11 computed from the Alexander polynomial of K as follows. Suppose ∆K (T ) = a0 + X ai (T i + T −i ). i>0 Define a sequence of integers ti = ∞ X jai+j , i ≥ 0, j=1 then ai can be recovered from ti by ai = ti−1 − 2ti + ti+1 , for i > 0. (3.6) If K admits an L-space surgery, then one can prove [21, 25] ts ≥ 0, ts ≥ ts+1 ≥ ts − 1, tg(K) = 0. (3.7) Moreover, the following proposition from Ozsváth and Szabó [21] and Rasmussen [25] holds. Proposition 3.3.2. Suppose the p/q-surgery on K ⊂ S 3 is an L-space surgery, where p, q > 0. Then for any 0 ≤ i < p we have 3 d(SK (p/q), i) = d(L(p, q), i) − 2 max{tb i c , tb p+q−1−i c }. q q 12 Chapter 4 Proof Strategy Recall that φ : Z/pZ → Spinc (YK (p/q)) is the natural identification defined by Ozsváth and Szabó, T is the exterior of the right-hand trefoil, and T(p/q) is the manifold obtained by p/q-surgery on the right-hand trefoil. The following two lemmas are from [10]. Lemma 4.0.3. Suppose i is an integer satisfying 0 ≤ i < p + q, then J(σ([i])) is represented by p + q − 1 − i. ) for some positive Lemma 4.0.4. Any T-type manifold is homeomorphic to ±T( 6q±3 q integer q with (q, 3) = 1. Any O-type manifold is homeomorphic to ±T( 6q±4 ) for q some positive integer q with (q, 2) = 1. Any I-type manifold is homeomorphic to ±T( 6q±5 ) for some positive integer q with (q, 5) = 1. q Let p, q > 0 be coprime integers. Using Proposition 3.3.2, we get d(T(p/q), i) = d(L(p, q), i) − 2χ[0,q) (i), where χ[0,q) (i) =   1 when 0 ≤ i < q  0 when q ≤ i < p. 3 Suppose SK (p) is a spherical space form, then by Proposition 3.3.2, 3 d(SK (p), i) = d(L(p, 1), i) − 2 max{ti , tp−i } = (2i − p)2 − p − 2tmin(i,p−i) . 4p (4.1) 13 3 If SK (p) ∼ = εT(p/q), ε = ±1, then the two sets 3 (p), i)|i ∈ Z/pZ}, {d(SK {εd(T(p/q), i)|i ∈ Z/pZ} are equal. However, the two parametrizations of Spinc structures may differ by an affine isomorphism of Z/pZ. More precisely, there exists an affine isomorphism φ : Z/pZ → Z/pZ, such that 3 d(SK (p), i) = εd(T(p/q), φ(i)). For any integers a, b, define φa,b : Z/pZ → Z/pZ by φa,b (i) = ai + b mod p. , j = 0, 1. Lemma 4.0.5. There are at most two values for b, bj = jp+q−1 2 Proof. The affine isomorphism φ commutes with J, i.e., φJp = J pq φ. Using lemma 4.0.3, we get the desired values for b. Note that b0 or b1 may be a half-integer, in this case we discard it. Note φa,b (i) = φp−a,b (p − i), By (3.1) and Lemma 4.0.3, d(T(p/q), φa,b (i)) = d(T(p/q), φp−a,b (p − i)) = d(T(p/q), φp−a,b (i)). So we may assume p 0<a< , 2 (p, a) = 1. (4.2) Then we may assume 3 d(SK (p), i) = εd(T(p/q), φa,bj (i)), for some a, any i ∈ Z/pZ, and j = 0 or 1. Let ∆εa,bj (i) = d(L(p, 1), i) − εd(T(p/q), φa,bj (i)). (4.3) By Proposition 3.3.2, we should have ∆εa,bj (i) = 2tmin(i,p−i) (4.4) 14 3 if SK (p) ∼ = εT(p/q) and φa,bj identifies their Spinc structures. In order to prove Theorem 1.0.1, we will compute the correction terms of the T-, O-, and I-type manifolds using (4.1). For all a satisfying (4.2), we compute the sequences ∆εa,bj (i). Then we check whether they satisfy (4.4) for some {ts } as in (3.7). We will show that (4.4) cannot be satisfied when p is sufficiently large. For small p, a direct computation yields all the p/q’s. By a standard argument in Heegaard Floer homology [20], we can get the knot Floer homology of the corresponding knots, which should be the knot Floer homology of either a (p, q)-torus knot ((p, q) = (2, 3), (2, 5), (3, 4), (3, 5)), a cable knot or some hyperbolic knot. We will list torus knots and cables of torus knots separately for completeness, one may also consult Moser [14] and Bleiler and Hodgson [3]. 15 Chapter 5 Obstructions When p Is Large 3 In this chapter, we will assume that SK (p) ∼ = εT(p/q), and p = 6q + ζr, r ∈ {3, 4, 5}, ε, ζ ∈ {−1, 1}. We will prove that this cannot happen when p is sufficiently large: 3 (p) ∼ 6= εT(p/q), where Proposition 5.0.6. If p > 310r(36r + 1)2 , then SK p = 6q + ζr, r ∈ {3, 4, 5}. Let s ∈ {0, 1, ..., r − 1} be the reduction of q modulo r. For any integer n, let θ(n) ∈ {0, 1} be the reduction of n modulo 2, and let θ̄(n) = 1 − θ(n). Lemma 5.0.7. For 0 ≤ i < q, 1−ζ q + ζr), i) = ζ d(L(q, 2   (2i + 1 − q − ζr)2 1 − − d(L(r, s), i mod r) . (5.1) 4qr 4 Proof. For 0 ≤ i < q, using (3.2), we have (2i + 1 − q − r)2 1 − − d(r, s, i mod r) 4qr 4 2 (2i + 1 − 2q + r) 1 d(L(q, q − r), i) = − − d(L(q − r, r), i) 4q(q − r) 4 2 (2i + 1 − 2q + r) (2i + 1 − q)2 = − + d(L(r, s), i mod r) 4q(q − r) 4r(q − r)   (2i + 1 − q + r)2 1 =− − − d(L(r, s), i mod r) . 4qr 4 d(L(q, r), i) = 16 Recall φa,b : Z/pZ → Z/pZ is defined by φa,b (i) = ai + b mod p. Lemma 5.0.8. When p > 52, there is at most one value for b in {b0 , b1 }. 3 Proof. For a T- or I-type p-surgery on a knot, by lemma 4.0.4, SK (p) ∼ = εT(p/q), where p = 6q + ζr, r = 3 or 5. Here p is odd, p2 is a half integer. By lemma 4.0.5, if q is odd, b = q−1 ; if q is even, b = p+q−1 . We may write b = θ̄(q)p+q−1 . 2 2 2 For an O-type p-surgery on a knot, by lemma 4.0.4, S 3 (p) ∼ = εT(p/q), where K p = 6q + ζr, r = 4. Note here p is even, (p, q) = (p, a) = 1, q, a are odd, so q = 4l + s, , j = 0, 1, and both of them are integers. where s = 1, 3. By lemma 4.0.5, bj = jp+q−1 2 Denote φa,j (i) = ai + bj . 3 More specifically, p = 6q + ζr, r = 4, SK (p) ∼ = εT(p/q), q = 4l + s, ζ, ε ∈ {1, −1}, s ∈ {1, 3}. For φa,j , φa,j (0) = jp+q−1 , φa,j ( p2 ) = (1−j)p+q−1 . 2 2 Using (3.2) and (5.1), we get p p d(L(p, 1), 0) − d(L(p, 1), ) = 2 4 q−1 p+q−1 d(L(p, q), ) − d(L(p, q), ) 2 2 1−ζ q−1 1−ζ q + ζr − 1 p2 − d(L(q, q + ζr), ) + d(L(q, q + ζr), ) = 4pq 2 2 2 2 p2 r2 s−1 s−1 = − ζ[ − d(L(4, s), 2l + mod 4) + d(L(4, s), 2l + 2 + 4pq 4qr 2 2 3 = + ζ(−1)l . 2 mod 4)] Here we require q − r > r and q+r−1 < q, it suffices to take p > 52 = 6 ∗ 2r + r. 2 Using Proposition 3.3.2, (4.1) and (4.4), we get p p 3 3 ∆εa,bj (0) − ∆εa,bj ( ) = ∓ ε( + ζ(−1)l − 2) = 6l + ζ ∓ εζ(−1)l + (s ∓ ε) ± 2ε. 2 4 2 2 The parity of ∆εa,bj (0) − ∆εa,bj ( p2 ) depends only on the parity of 32 (s ∓ ε), and by 17 (4.4), it should be even, so we get b=   q−1  We can write b = 2 p+q−1 2 if s = 1, ε = 1 or s = 3, ε = −1 . if s = 1, ε = −1 or s = 3, ε = 1 θ̄( s+ε )p+q−1 2 for p > 52. 2 Because of Lemma 5.0.8, we can treat T-, O-, and I-type manifolds uniformly. Let θ = θ(q, ε) =   θ̄(q) if r = 3, 5  θ̄( s+ε ) if r = 4, q = 4l + s 2 , , we may denote φa,b by φa,θ . then b = θp+q−1 2 3 Lemma 5.0.9. Assume that SK (p) ∼ = εT(p/q). Let m ∈ {0, 1, 2, 3} satisfy that 0 ≤ a − mq + θζr + q − 1 < q, 2 then mp |< |a − 6 r 11rp . 6   0 if 0 ≤ θp+q−1 + a < p 2 ε ε Proof. By (3.7), ∆a,θ (0) − ∆a,θ (1) = 0 or 2. Let h = .  1 if θp+q−1 + a ≥ p 2 ∆εa,θ (0) − ∆εa,θ (1) (5.2) θp + q − 1 θp + q − 1 ) − 2χ[0,q) ( )] 2 2 θp + q − 1 θp + q − 1 − d(L(p, 1), 1) + ε[d(L(p, q), + a) − 2χ[0,q) ( + a − hp)] 2 2 θp + q − 1 θp + q − 1 p2 (p − 2)2 =2ε{χ[0,q) ( ) − χ[0,q) ( + a − hp)} + − 2 2 4p 4p 2 [(θ − 1)p] − pq 1−ζ θζr + q − 1 − ε{ − d(L(q, q + ζr), ) 4pq 2 2 [2a + (θ − 1)p]2 − pq 1−ζ θζr + q − 1 − + d(L(q, q + ζr), + a − mq)} 4pq 2 2 =d(L(p, 1), 0) − ε[d(L(p, q), Let i = θζr+q−1 mod r, j = θζr+q−1 + a − mq mod r. 2 2 18 Since 0 ≤ θζr+q−1 + a − mq < q, we use (5.1), the right-hand side of (5.2) becomes 2   θp + q − 1 θp + q − 1 p−1 a[a + (θ − 1)p] 2ε χ[0,q) ( ) − χ[0,q) ( + a − hp) + +ε + 2 2 p pq   [(θ − 1)ζr]2 − qr [2a − 2mq + (θ − 1)ζr]2 − qr εζ − d(L(r, s), i) − + d(L(r, s), j) 4qr 4qr [a − mq + (θ − 1)ζr](a − mq) a[a + (θ − 1)p] − εζ =C +ε pq qr 2 6εζ mp 2 εm =− (a − ) − εm(1 − θ) + + C, pr 6 6 where θp + q − 1 θp + q − 1 ) − χ[0,q) ( + a − hp)} 2 2 p−1 . + εζ[d(L(r, s), j) − d(L(r, s), i)] + p C =2ε{χ[0,q) ( Using (2),(3),(4), |C| ≤ 65 + 2 + 1 < 29 . 2 2 Moreover, | − εm(1 − θ) + εm6 | ≤ m + m6 ≤ 3 + 32 = 29 . So we get | 6 mp 2 9 9 (a − ) | < 2 + + = 11, pr 6 2 2 so our conclusion holds. Lemma 5.0.10. Suppose p > 767. Let k be an integer satisfying 1 0≤k< 6 ! √ 6 √ √ p−1 . 13 11r (5.3) Let ik = θζr + q − 1 θζr + q − 1 +6ka−kmp mod r, jk = +(6k+1)a−kmp−mq 2 2 mod r. 19 Then ∆εa,θ (6k) − ∆εa,θ (6k + 1) = Ak + B + Ck , where mp 2 12 72εζ (a − ) − , A=− pr 6 p   6ζ mp 2 m2 B = ε − (a − ) − m(1 − θ) + pr 6 6 + 2ε{2ε{χ[0,q) (3θq) − χ[0,q) ((3θ + m − 6h)q)} + p−1 , p Ck = εζ[d(L(r, s), jk ) − d(L(r, s), ik )]. and   0 if 0 ≤ 3θ + m < 6 h= .  1 if 3θ + m = 6 Proof. Using(4.3), we get ∆εa,θ (6k) − ∆εa,θ (6k + 1) (5.4) θp + q − 1 + 6ka − kmp) 2 θp + q − 1 θp + q − 1 − 2χ[0,q) ( + 6ka − kmp)] + ε[d(L(p, q), + (6k + 1)a − kmp) 2 2 θp + q − 1 + (6k + 1)a − (km + h)p)] − 2χ[0,q) ( 2 θp + q − 1 θp + q − 1 =2ε{χ[0,q) ( + 6ka − kmp) − χ[0,q) ( + (6k + 1)a − (km + h)p)} 2 2 (p − 12k)2 [p − 2(6k + 1)]2 + − 4p 4p [12ka − (2km + 1 − θ)p]2 [(12k + 2)a − (2km + 1 − θ)p]2 − ε{ − 4pq 4pq 1−ζ θζr + q − 1 − d(L(q, q + ζr), + 6ka − kmp) 2 2 1−ζ θζr + q − 1 + d(L(q, q + ζr), + (6k + 1)a − kmp − mq)}. 2 2 =d(L(p, 1), 6k) − d(L(p, 1), 6k + 1) − ε[d(L(p, q), We require θζr + q − 1 + 6ka − kmp < q, 2 θζr + q − 1 0≤ + (6k + 1)a − kmp − mq < q. 2 0≤ 20 It suffices that 1 k< 6  q−9 2 r  6 −1 . 11rp This implies θp + q − 1 + 6ka − kmp < (3θ + 1)q, 2 θp + q − 1 (3θ + m)q ≤ + (6k + 1)a − kmp < (3θ + m + 1)q. 2 3θq ≤ When m = 3, θ = 1, (5.6) becomes 6q ≤ θp + q − 1 + (6k + 1)a − kmp < 7q. 2 Here we require p≤ θp + q − 1 + (6k + 1)a − kmp < p + q. 2 We know a < p2 , so θp + q − 1 p θp + q − 1 + (6k + 1)a − kmp < + < p + q. 2 2 2 q Moreover, we know when m = 3, by Lemma 5.0.9, a > p2 − 11rp . 6 If 1 k≤ 6  q−1 2 r  6 −1 , 11rp then θp + q − 1 + (6k + 1)a − kmp ≥ p. 2 When p > 767, 1 6  p 13 r r    6 1 q−9 6 −1 < −1 . 11rp 6 2 11rp (5.5) (5.6) 21 Using (5.1), (5.5) and (5.6), the right-hand side of (5.4) becomes 2ε{χ[0,q) (3θq) − χ[0,q) ((3θ + m − 6h)q)} + p − (12k + 1) p a((12k + 1)a − 2kmp + (θ − 1)p) pq [12ka − 2kmp + (θ − 1)ζr]2 − qr + εζ{ − d(L(r, s), ik ) 4qr [2(6k + 1)a − 2kmp − 2mq + (θ − 1)ζr]2 − qr − + d(L(r, s), jk )}. 4qr +ε This simplifies to be 2ε{χ[0,q) (3θq) − χ[0,q) ((3θ + m − 6h)q)} + εζ[d(L(r, s), jk ) − d(L(r, s), ik )] p − (12k + 1) a((12k + 1)a − 2kmp + (θ − 1)p) +ε p pq ((12k + 1)a − 2kmp − mq + (θ − 1)ζr)(a − mq) − εζ qr mp 2 εm2 6(12k + 1)εζ (a − ) − εm(1 − θ) + =− pr 6 6 + + 2ε{χ[0,q) (3θq) − χ[0,q) ((3θ + m − 6h)q)} + εζ[d(L(r, s), jk ) − d(L(r, s), ik )] + p − (12k + 1) p =Ak + B + Ck . 3 Proof of Proposition 4.0.8. If SK (p) ∼ = εT(p/q), then (4.4) holds, so ∆εa,θ (6k) − ∆εa,θ (6k + 1) = 0 or 2 (5.7) for all k satisfying (5.3). If p > 310r(36r + 1)2 , then √ 6 √ 6 · 6r + 1 < √ p 13 11r hence k = 6r satisfies (5.3). Let A, B, Ck be as in Lemma 4.4. If A 6= 0, then Ak + B + C is equal to 0 or 2 for at most two values of k for any given C. Given p, q, a, ε, ζ, as k varies, Ck can take 22 at most 3r values. It follows that Ak + B + Ck cannot be 0 or 2 for k = 0, 1, ..., 6r. As a consequence, if p > 310r(36r + 1)2 , then (5.7) does not hold. The only case we need to consider is that A = 0. In this case εζ = −1. 12 A= p We get |a − mp |= 6 pr 6   6 mp 2 (a − ) −1 =0 r 6 , which is an irrational number. This contradicts that a is an integer and mp is a rational number. 6 Since we get an upper bound for p, an easy computer search will yield all possible p/q’s. They are 1/1, 2/1, 3/1, 7/2, 9/1, 9/2, 10/1, 10/1, 11/1, 13/3, 13/3, 14/3, 17/2, 17/2, 19/4, 21/4, 22/3, 23/3, 27/4, 27/5, 29/4, 29/4, 37/7, 37/7, 38/7, 43/8, 46/7, 47/7, 49/9, 50/9, 51/8, 58/9, 59/9, 62/11, 69/11, 70/11, 81/13, 81/14, 83/13, 86/15, 91/16, 93/16, 94/15, 99/16, 99/17, 101/16, 106/17, 106/17, 110/19, 110/19, 113/18, 113/18, 119/19, 131/21, 133/23, 137/22, 137/22, 143/23, 146/25, 154/25, 157/27, 157/27, 163/28, 211/36, 221/36. Here if pi /qi appears twice, this means they correspond to candidate knots with different Heegaard Floer Homologies. 23 Chapter 6 Constructions When p Is Small 6.1 Dean’s Construction Let H be a genus two handlebody, and α a simple closed curve lying on ∂H. Denote by H(α) the manifold obtained by adding a 2-handle to H along the curve α. We say that α is primitive with respect to H if H(α) is a solid torus. We say that α is (m, n) Seifert-fibered if H(α) is a Seifert-fibered manifold over the disk with two exceptional fibers of multiplicities m and n. Equivalently, we say that α is primitive with respect to H if it is part of a basis for the free group π1 (H), and α is (m, n) Seifert-fibered if it represents a word w in the free group π1 (H) =< x, y >, such that < x, y|w >∼ =< a, b|am bn > for some nonzero integers m and n. Let K be a knot contained in a genus two Heegaard surface Σ for S 3 , that is, S 3 = H ∪Σ H 0 , where H and H 0 are genus two handlebodies. Let r be the surface 3 (r) = H(r) ∪Σ0 H 0 (r). If K slope of K with respect to Σ. Dean [6] proved that SK 3 is primitive/primitive, then SK (r) is a lens space. If K is primitive/Seifert-fibered, 3 (r) is a Seifert-fibered manifold over the sphere with at most three exceptional then SK fibers or a connected sum of two lens spaces. Let ∆(r1 , r2 ) be the minimal geometric intersection number between two slopes r1 and r2 . Theorem 6.1.1. (Boyer and Zhang [4]) Let K ⊂ S 3 be a hyperbolic knot. If r1 is a finite surgery slope and r2 is a cyclic surgery slope of K, then ∆(r1 , r2 ) ≤ 2. This motivates the following results: 24 Proposition 6.1.2. Only 11 hyperbolic Berge knots have T-, O-, and I-type surgeries. More precisely, let K(p, q; ω) be the Berge knot corresponding to homology class ω in L(p, q). They are K(18, 5; 5), K(39, 16; 16), K(45, 19; 8), K(46, 19; 11), K(68, 19; 5), K(71, 27; 11), K(82, 23; 5), K(93, 26; 5), K(107, 30; 5), K(118, 33; 5), K(132, 37; 5). Proof. Berge knots have lens space surgeries, we can compute their Heegaard Floer Homology using Proposition 3.3.2 and compare them with the list from finite surgeries, we get 11 candidates. We draw the link diagram in SnapPy and compute the fundamental group of the Dehn filling with the finite surgery coefficient. If what we get is a (2, 3, n)-type group, then we have verified that the knot has indeed a required finite surgery. Below is a table of the candidates. Table 6.1: Candidates of Berge knots with T-, O-, and I-type surgeries p q ω finite surgery coefficient p’ 18 5 5 17 39 16 16 38 45 19 8 46 46 17 11 47 68 19 5 69 71 21 11 70 82 23 5 81 93 25 5 94 107 25 5 106 118 25 5 119 132 25 5 131 Here we use a point of view of dual Berge knots in the corresponding lens space, as they have the same knot complement as Berge knots in S 3 , the only thing is to figure out the corresponding coefficients. The computation is as follows. 25 0 1 Figure 6.1: Link for K(107, 30; 5) We would like to draw a link L with two components as above. When we perform on the trivial component (1-component) p/q-surgery, we get L(p, q). The other component then becomes the dual Berge knot in L(p, q). We would like to choose the orientations of the two components consistently, and one choice is shown above. Denote the longitude and meridian of 0-component and 1-component by λ and µ, and l and m, respectively. In homology, we have ωµ = l and λ = ωm. After performing p/q-surgery on 1-component, we have pm + ql = 0 in homology. The resultant 0-component K 0 is the dual Berge knot, and its longitude and meridian are denoted by λ0 and µ0 respectively. We would like to write λ0 and µ0 in terms of λ and µ. By performing p-surgery on K 0 we get L(p, q), and by performing ∞-surgery on K 0 we get S 3 . Note we have qω 2 µ + pλ = qωl + pωm = 0, this means λ0 = (qω 2 , p). Note that L(p, q1 ) ∼ = L(p, q2 ) if and only if q1 q2 ≡ ±1 (mod p), so there is an indeterminacy in q. Take K(107, 25; 5) for example. We have λ0 = (25 ∗ 52 , 107) and 107µ0 + λ0 = (±1, 0), there is no integer solutions. As 30 ∗ 25 ≡ 1 (mod 107), we take q = 30 and denote the knot by K(107, 30; 5). We have λ0 = (30∗52 , 107) and 107µ0 +λ0 = (±1, 0), so we get µ0 = (−7, −1), 107µ0 + λ0 = (1, 0) and 106µ0 + λ0 = (8, 1). Now we have S 3 (1/0, 107/30) ∼ = L(107, 30) and S 3 (7/1, 107/30) ∼ = S 3 , we hope to L L get a spherical space form by performing p0 -surgery on K 0 . By SnapPy, SL3 (8/1, 107/30) has fundamental group of the form < a, b|abab−2 , ab(b3 a2 )77 b4 ab((a−2 b−3 )3 a−1 )21 a−1 b >. 26 The first relator is (ab)2 = b3 , which is in the center. Mod it out, the other relator becomes a4 = 1. By upper central series theory, this group is finite and is of type (2, 3, 4), which means this dehn filling is indeed the required finite surgery. Table 6.2: Berge knots with T-, O-, and I-type surgeries knot p0 3 Z(π1 (SK (p0 ))) 3 3 π1 (SK (p0 ))/Z(π1 (SK (p0 ))) K(18, 5; 5) 17 h(ab)2 i ha, b|(ab)2 , b3 , a5 i K(39, 16; 16) 38 ha4 i ha, b|b2 , (a−1 b)3 , a4 i K(45, 19; 8) 46 h(ab)2 i ha, b|(ab)2 , a3 , b4 i K(46, 17; 11) 47 hb3 i h(b2 a−2 )2 , b3 , a5 i K(68, 19; 5) 69 h(ab)2 i ha, b|(ab)2 , b3 , a3 i K(71, 27; 11) 70 ha3 i ha, b|b2 , a3 , (ba2 )4 i K(82, 23; 5) 81 h(ab)2 i ha, b|(ab)2 , b3 , a3 i K(93, 26; 5) 94 h(ab)2 i ha, b|(ab)2 , b3 , a4 i K(107, 30; 5) 106 h(ab)2 i ha, b|(ab)2 , b3 , a4 i K(118, 33; 5) 119 h(ab)2 i ha, b|(ab)2 , b3 , a5 i K(132, 37; 5) 131 h(ab)2 i ha, b|(ab)2 , b3 , a5 i Many examples of primitive/Seifert-fibered knots can be found among the twisted torus knots, which Dean defined in [6]. Let V be a standardly embedded solid torus in B 3 , the 3-ball. ∂B 3 intersects with ∂V in a 2-disk D. Let T (p, q) be a (p, q) torus knot on ∂V , and T (p, q) intersects r times with D. Remove T (p, q)∩D from T (p, q) we get a torus tangle, denoted by T (p, q)r , on a once-punctured torus ∂V \D. Similarly, we can get T (m, n)1 on a once-punctured torus ∂V 0 \D0 . Take r parallel copies of T (m, n)1 , denote by rT (m, n)1 , on ∂V 0 \D0 . Glue ∂V \D and ∂V 0 \D0 along the boundary, we get a knot T (p, q)r +rT (m, n)1 . The knot obtained in this manner is called a twisted torus knot, and is denoted by K(p, q, r, m, n), where (p, q) = (m, n) = 1, and 0 < r ≤ p + q. Not all twisted torus knots are primitive/Seifert-fibered. One necessary condition for the knot to be primitive is m = 1 if p 6= 1. From now on, we will denote K(p, q, r, 1, n) by K(p, q, r, n). Dean [6] gave two special classes of twist torus knots which are primitive/Seifert-fibered: 27 Theorem 6.1.3. (Dean [6]) 1. (pq + (p − kq)2 )-surgery on K(p, q, p − kq, 1) produces a Seifert-fibered manifold with base orbifold S 2 (µ1 , µ2 , µ3 ) = S 2 (p + q − kq, p − kq, k) with (µ1 , µ2 ) = 1 and |µ1 − µ2 | > 1. 2. (pq − (p − kq)2 )-surgery on K(p, q, p−kq, −1) produces a Seifert-fibered manifold with base orbifold S 2 (µ1 , µ2 , µ3 ) = S 2 (−p + q + kq, p − kq, k). Miyazaki and Motegi [12] gave another special class of twist torus knots which are primitive/Seifert-fibered: Proposition 6.1.4. (Miyazaki and Motegi [12]) (pq + (p + q)2 n)-surgery on K(p, q, p+ q, n) produces a Seifert-fibered manifold with base orbifold S 2 (|p|, q, |n|). 6.2 Montesinos Construction An n-tangle (B 3 , t) is a pair that consists of a 3-ball B 3 and n disjoint arcs t properly embedded in B 3 , and possibly some simple closed curves. (B 3 , t) is a trivial tangle if there is an homeomorphism of pairs from (B 3 , t) to the tangle (D2 ×I, {x1 , ..., xn }×I) where D2 is the unit disk in R2 and x1 , ..., xn are distinct points in the interior of D2 . A rational tangle is a trivial 2-tangle. There is a natural one to one correspondence between rational tangles and Q ∪ {1/0}. Denote by R(p/q) the rational tangle determined by p/q ∈ Q ∪ {1/0}. A Montesinos link, denoted by K(e; β1 /α1 , . . . , βn /αn ), is composed of n rational tangles. The double branched cover of S 3 , with branched set a Montesinos link of type (e; β1 /α1 , ..., βn /αn ), is a Seifert-fibred space over S 2 (α1 , ..., αn ), and that every surgery on a strongly invertible link yields a manifold which is a double branched cover of S 3 branched over a link in S 3 [13]. We first try to find a strongly invertible link L ⊂ S 3 , such that some surgery on L yields a manifold which is a double cover of S 3 , branched over a Montesinos link of type (e, β1 /α1 , β2 /α2 , β3 /α3 ) with (α1 , α2 , α3 ) = (2, 3, m) or (2, 2, n), where m = 3, 4, 5 and n ≥ 2. On the other hand, the surgery on L may be transferred, using Kirby calculus, to a surgery on a hyperbolic knot K ⊂ S 3. 28 6.3 Berge’s Construction Berge and Kang [2] claimed that they produced a complete list of the hyperbolic primitive/Seifert-fibered knots in S 3 . If their list is indeed complete, all knots in our list can be found in their list. Most of the work here was done before their paper, and we have taken four knots from their list. 29 Chapter 7 Results I summarize all results below with a table of torus knots and satellite knots and a table of hyperbolic knots. Let T (p, q) be the (p, q) torus knot, and let [p1 , q1 ; p2 , q2 ] denote the (p1 , q1 )-cable of T (p2 , q2 ). A knot K is said to be a (p, q)-cable of a knot k if K is a simple loop in the boundary of neighborhood N (k) wrapping p times in the meridional direction and q times in the longitudinal direction. I also include pictures of all the hyperbolic knots using Mathematica package KnotTheory`. Table 7.1: Torus knots and satellite knots with T-, O-, and I-type surgeries p knot p knot p knot p knot p knot 1 T (3, 2) 2 T (3, 2) 3 T (3, 2) 7 T (5, 2) 9 T (3, 2) 10 T (3, 2) 10 T (4, 3) 11 T (3, 2) 13 T (5, 3) 13 T (5, 2) 14 T (4, 3) 17 T (5, 3) 19 [9, 2; 3, 2] 21 [11, 2; 3, 2] 27 [13, 2; 3, 2] 29 [15, 2; 3, 2] 37 [19, 2; 5, 2] 43 [21, 2; 5, 2] 49 [16, 3; 3, 2] 50 [17, 3; 3, 2] 59 [20, 3; 3, 2] 91 [23, 4; 3, 2] 93 [23, 4; 3, 2] 99 [25, 4; 3, 2] 101 [25, 4; 3, 2] 106 [35, 3; 4, 3] 110 [37, 3; 4, 3] 133 [44, 3; 5, 3] 137 [46, 3; 5, 3] 146 [29, 5; 3, 2] 154 [31, 5; 3, 2] 157 [39, 4; 5, 2] 163 [41, 4; 5, 2] 211 [35, 6; 3, 2] 221 [37, 6; 3, 2] 30 Table 7.2: Hyperbolic knots with T-, O-, and I-type surgeries p knot 17 Pretzel knot P (−2, 3, 7) 22 Pretzel knot P (−2, 3, 9) 23 Pretzel knot P (−2, 3, 9) 29 mirror image of K(1, 1, 0) from section 4 of [8] 37 K(11, 3, 2, 1) from [6] 38 Berge knot K(39, 16; 16) 46 Berge knot K(45, 19; 8) 47 K3∗ from [4] 51 K2] from [4] 58 mirror image of the P/SFd KIST IV knot with (n, p, , J1 , J2 ) = (−2, 1, 1, −4, 1) from [2] 62 P/SFd KIST III knot with (h, k, h0 , k 0 , J) = (−5, −3, −2, −1, 1) from [2] 69 mirror image of K(2, 3, 5, −3) from [12] 70 Berge knot K(71, 27; 11) 81 K(2, 3, 5, 3) from [12] 83 P/SFd KIST V knot with (n, p, , J1 , J2 ) = (1, −2, −1, 2, 2) from [2] 86 mirror image of K(3, 4, 7, −2) from [12] 94 mirror image of K(2, 3, 5, −4) from [12] 106 K(2, 3, 5, 4) from [12] 110 K(3, 4, 7, 2) from [12] 113 mirror image of K(3, 5, 8, −2) from [12] 113 mirror image of the P/SFd KIST V knot with (n, p, , J1 , J2 ) = (−3, −2, −1, 2, 2) from [2] 119 mirror image of K(2, 3, 5, −5) from [12] 131 K(2, 3, 5, 5) from [12] 137 mirror image of K(2, 5, 7, −3) from [12] 143 K(3, 5, 8, 2) from [12] 157 K(2, 5, 7, 3) from [12] 31 Figure 7.1: Pretzel knot P (−2, 3, 7) Figure 7.2: Pretzel knot P (−2, 3, 9) 32 Figure 7.3: Mirror image of K(1, 1, 0) from section 4 of [8] Figure 7.4: Twisted torus knot K(11, 3, 2, 1) from [6] 33 Figure 7.5: Berge knot K(39, 16; 16) Figure 7.6: Berge knot K(45, 19; 8) 34 Figure 7.7: K3∗ from [4] Figure 7.8: K2] from [4] Figure 7.9: Mirror image of the P/SFd KIST IV knot with (n, p, , J1 , J2 ) = (−2, 1, 1, −4, 1) from [2] 35 Figure 7.10: P/SFd KIST III knot with (h, k, h0 , k 0 , J) = (−5, −3, −2, −1, 1) from [2] Figure 7.11: Mirror image of twisted torus knot K(2, 3, 5, −3) from [12] Figure 7.12: Berge knot K(71, 27; 11) 36 Figure 7.13: Twisted torus knot K(2, 3, 5, 3) from [12] Figure 7.14: P/SFd KIST V knot with (n, p, , J1 , J2 ) = (1, −2, −1, 2, 2) from [2] Figure 7.15: Mirror image of twisted torus knot K(3, 4, 7, −2) from [12] 37 Figure 7.16: Mirror image of twisted torus knot K(2, 3, 5, −4) from [12] Figure 7.17: Twisted torus knot K(2, 3, 5, 4) from [12] Figure 7.18: Twisted torus knot K(3, 4, 7, 2) from [12] 38 Figure 7.19: Mirror image of twisted torus knot K(3, 5, 8, −2) from [12] Figure 7.20: Mirror image of the P/SFd KIST V knot with (n, p, , J1 , J2 ) = (−3, −2, −1, 2, 2) from [2] Figure 7.21: Mirror image of twisted torus knot K(2, 3, 5, −5) from [12] Figure 7.22: Twisted torus knot K(2, 3, 5, 5) from [12] 39 Figure 7.23: Mirror image of twisted torus knot K(2, 5, 7, −3) from [12] Figure 7.24: Twisted torus knot K(3, 5, 8, 2) from [12] Figure 7.25: Twisted torus knot K(2, 5, 7, 3) from [12] 40 Bibliography [1] J. 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