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On Cosmetic Surgery on Knots K Cosmetic surgery on knots On cosmetic surgery on knots K. Ichihara Introduction 3-manifold Dehn surgery Cosmetic Surgery Kazuhiro Ichihara Conjecture Known Facts Examples Nihon University Criteria College of Humanities and Sciences Results (1) Jones polynomial 2-bridge knot Hanselman's result Finite type invariants Based on Joint work with Recent progress Tetsuya Ito (Kyoto Univ.), In Dae Jong (Kindai Univ.), Results (2) Genus one alternating knots Thomas Mattman (CSU, Chico), Toshio Saito (Joetsu Univ. of Edu.), Cosmetic Banding and Zhongtao Wu (CUHK) How To Check? The 67th Topology Symposium - Online, MSJ. Oct 13, 2020. 1 / 36 Cosmetic surgery Papers on knots I (with Toshio Saito) K. Ichihara Cosmetic surgery and the SL(2; C) Casson invariant for 2-bridge knots. Introduction 3-manifold Hiroshima Math. J. 48 (2018), no. 1, 21-37. Dehn surgery Cosmetic Surgery I (with In Dae Jong (Appendix by Hidetoshi Masai)) Conjecture Known Facts Cosmetic banding on knots and links. Examples Osaka J. Math. 55 (2018), no. 4, 731-745. Criteria Results (1) I (with Zhongtao Wu) Jones polynomial 2-bridge knot A note on Jones polynomial and cosmetic surgery. Hanselman's result Finite type invariants Comm. Anal. Geom. 27 (2019), no.5, 1087{1104. Recent progress I (with Toshio Saito and Tetsuya Ito) Results (2) Genus one alternating Chirally cosmetic surgeries and Casson invariants. knots Cosmetic Banding To appear in Tokyo J. Math. How To Check? I (with In Dae Jong, Thomas W. Mattman, Toshio Saito) Two-bridge knots admit no purely cosmetic surgeries. To appear in Algebr. Geom. Topol. 2 / 36 Table of contents Introduction Jones polynomial 3-manifold 2-bridge knot Dehn surgery Hanselman's result Finite type invariants Cosmetic Surgery Conjecture Recent progress Known Facts Results (2) [Chirally cosmetic surgery] Examples Alternating knot of genus one Criteria Cosmetic Banding Results (1) [Purely cosmetic surgery] How To Check? Cosmetic surgery Classification of 3-manifolds on knots K. Ichihara FACT Introduction 3-manifold Every closed orientable 3-manifold is; Dehn surgery Cosmetic Surgery Conjecture I Reducible (containing essential sphere) Known Facts Examples I Toroidal (containing essential torus) Criteria Results (1) I Seifert fibered (admitting a foliation by circles) Jones polynomial 2-bridge knot Hanselman's result I Hyperbolic (admitting Riem.metric of const.curv.−1) Finite type invariants Recent progress Results (2) Genus one alternating as a consequence of the Geometrization Conjecture knots Cosmetic Banding including famous Poincar´eConjecture (1904) How To Check? conjectured by Thurston (late '70s) established by Perelman (2002-03) 4 / 36 Cosmetic surgery What's NEXT? on knots K. Ichihara Introduction 3-manifold I Dehn surgery Attack the remaining Open Problems. Cosmetic Surgery Conjecture (e.g., \Heegaard genus VS rank of π1" problem, ::: ) Known Facts Examples Criteria I Relate Geometric & Topological invariants. (e.g., Volume conjecture ::: ) Results (1) Jones polynomial 2-bridge knot Hanselman's result I ······ Finite type invariants Recent progress I Results (2) Study the Relationships between 3-manifolds. (e.g., Dehn surgery ::: ) Genus one alternating knots (* Today!) Cosmetic Banding How To Check? 5 / 36 Cosmetic surgery Dehn surgery on a knot on knots K : a knot (i.e., embedded circle) in a 3-manifold M K. Ichihara Introduction Dehn surgery on K 3-manifold Dehn surgery Cosmetic Surgery 1) remove the open neighborhood of K from M (to obtain the exterior E(K) of K) Conjecture Known Facts 2) glue a solid torus back (along a slope γ) Examples Criteria Results (1) Jones polynomial 2-bridge knot Hanselman's result f Finite type invariants Recent progress Results (2) γ m Genus one alternating knots Cosmetic Banding We denote the obtained manifold by K(γ). How To Check? a slope := an isotopy class of unoriented, non-trivial simple closed curve on T 2 Slopes are parametrized by Q [ f 1=0 g w.r.t. a meridian-logitude system 6 / 36 Cosmetic surgery Cosmetic surgery on knots K. Ichihara It is natural to ask: Introduction 3-manifold Can a pair of distinct Dehn surgeries give the same manifold? Dehn surgery Cosmetic Surgery Conjecture Known Facts Examples Cosmetic Surgery Conjecture [Gordon, '90], [Kirby's list, Problem 1.81(A)] Criteria Results (1) A pair of Dehn surgeries on inequivalent slopes are never purely cosmetic. Jones polynomial 2-bridge knot Hanselman's result I A pair of slopes are equivalent if there exists a homeomorphism of E(K) Finite type invariants Recent progress taking one slope to the other. Results (2) Genus one alternating I knots A pair of surgeries on K along slopes r1; r2 are purely cosmetic Cosmetic Banding How To Check? if there exists an ori.-pres. homeo. between K(r1) & K(r2), and chirally cosmetic if the homeo. is orientation-reversing. 7 / 36 Table of contents Introduction Jones polynomial 3-manifold 2-bridge knot Dehn surgery Hanselman's result Finite type invariants Cosmetic Surgery Conjecture Recent progress Known Facts Results (2) [Chirally cosmetic surgery] Examples Alternating knot of genus one Criteria Cosmetic Banding Results (1) [Purely cosmetic surgery] How To Check? Cosmetic surgery Known facts (Examples) on knots Mathieu (1992) K. Ichihara 18k+9 18k+9 3 Introduction The 3k+1 - and 3k+2 -surgeries on the trefoil knot T2;3 in S yield 3-manifold Dehn surgery orientation-reversingly homeomorphic pairs for any k ≥ 0. Cosmetic Surgery Conjecture Known Facts Rong (1995) Examples Criteria Seifert knots in closed 3-manifolds (except lens spaces) admitting cosmetic Results (1) Jones polynomial surgeries are classified. 2-bridge knot Hanselman's result Finite type invariants As a corollary, we have the following: Recent progress 3 Let Tr;s be the (r; s)-torus knot in S . For a positive integer m, Results (2) 2r2(2m+1) ∼ 2r2(2m+1) Genus one alternating T ( ) = −T ( ) if and only if r ≥ 3, odd, s = 2. knots r;s r(2m+1)+1 r;s r(2m+1)−1 Cosmetic Banding How To Check? Matignion (2010) Non-hyperbolic knots in lens spaces admitting cosmetic surgeries are classified. Remark: These surgeries are all chirally cosmetic. 9 / 36 Cosmetic surgery Known facts (Examples) on knots Bleiler-Hodgson-Weeks (1999) K. Ichihara Introduction There exists a hyperbolic knot which admits a pair of surgeries 3-manifold Dehn surgery along inequivalent slopes yielding oppositely oriented lens spaces. Cosmetic Surgery Conjecture Known Facts Examples The lens spaces are: L(49; −19) $ L(49; −18) (mirror image) Criteria Results (1) Jones polynomial 2-bridge knot Hanselman's result Finite type invariants Recent progress Results (2) Genus one alternating knots Cosmetic Banding How To Check? 10 / 36 Cosmetic surgery Known facts (Criteria) on knots K. Ichihara Introduction 3 3-manifold Let ∆K (t) denote the Alexander polynomial of a knot K in S Dehn surgery Cosmetic Surgery normalized to be symmetric and satisfy ∆K (1) = 1. Conjecture Known Facts Examples Criteria Boyer-Lines (1990) Results (1) 00 Jones polynomial 6 2-bridge knot A knot K satisfying ∆K (1) = 0 has no cosmetic surgeries. Hanselman's result Finite type invariants Recent progress They use the Casson invariant (defined by using SU(2)-representations). Results (2) Genus one alternating knots 00 1 Cosmetic Banding Remark: ∆K (1) = 2 a2(K) (the 2nd coefficient of the Conway polynomial of K) How To Check? 11 / 36 Cosmetic surgery Known facts (Criteria) on knots K. Ichihara Based on Heegaard Floer homology theory , developed by Ozsv´athand Szab´o, Introduction the following are obtained. 3-manifold Dehn surgery Cosmetic Surgery Theorem [Ni-Wu (2011)] Conjecture Known Facts 3 Suppose K is a nontrivial knot in S & r1; r2 are distinct slopes Examples ∼ Criteria such that K(r1) = K(r2) as oriented manifolds. Results (1) Jones polynomial Then r ; r satisfy that 2-bridge knot 1 2 Hanselman's result (a) r1 = −r2 Finite type invariants 2 Recent progress (b) suppose r1 = p=q, then q ≡ −1 (mod p) Results (2) (c) τ(K) = 0, where τ is the invariant defined by Ozsv´ath-Szab´o. Genus one alternating knots Cosmetic Banding How To Check? They use Heegaard Floer d-invariant and τ-invariant. Remark: If K is alternating, τ(K) = σ(K) (signature of K) 12 / 36 Table of contents Introduction Jones polynomial 3-manifold 2-bridge knot Dehn surgery Hanselman's result Finite type invariants Cosmetic Surgery Conjecture Recent progress Known Facts Results (2) [Chirally cosmetic surgery] Examples Alternating knot of genus one Criteria Cosmetic Banding Results (1) [Purely cosmetic surgery] How To Check? Cosmetic surgery Jones polynomial on knots 3 Next theorem gives a severe restriction for a knot in S K. Ichihara to admit purely cosmetic surgeries in terms of its Jones polynomial. Introduction 3-manifold Dehn surgery Cosmetic Surgery Theorem [I.-Wu, 2019] Conjecture Known Facts 3 Examples Let VK (t) be the Jones polynomial of a knot K in S . Criteria 00 000 If a knot K satisfies either V (1) =6 0 or V (1) =6 0 , Results (1) K K Jones polynomial ≇ 0 0 2-bridge knot then K(r) K(r ) as oriented mfds. for distinct slopes r and r . Hanselman's result Finite type invariants Recent progress Remark Results (2) Genus one alternating Boyer and Lines obtained a similar result for a knot K knots 00 6 Cosmetic Banding with ∆K (1) = 0 by using the Casson invariant. How To Check? (∆K (t): the normalized Alexander polynomial) 00 − 00 Since VK (1) = 3∆K (1), our result can be viewed as an extension of [Proposition 5.1, Boyer-Lines (1990)]. 14 / 36 Cosmetic surgery Lescop's λ2 invariant on knots The essential new ingredient for our result is the following. K. Ichihara Introduction 3-manifold Dehn surgery Lescop's λ invariant (2009) Cosmetic Surgery 2 Conjecture The invariant λ := W ◦ Z , where W is a linear form on A Known Facts 2 2 2 2 n Examples Criteria with W2( ) = 1 and W2( ) = 0.
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