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 ▶ (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 ▶ (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) ▶ (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 ▶ (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? ▶ (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

K. Ichihara

FACT Introduction 3-manifold Every closed orientable 3-manifold is; Dehn surgery Cosmetic Surgery Conjecture ▶ Reducible (containing essential sphere) Known Facts Examples ▶ Toroidal (containing essential torus) Criteria Results (1) ▶ Seifert ﬁbered (admitting a foliation by circles) Jones polynomial 2-bridge knot Hanselman’s result ▶ 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 ▶ Dehn surgery Attack the remaining Open Problems. Cosmetic Surgery Conjecture (e.g., “Heegaard genus VS rank of π1” problem, ... ) Known Facts Examples Criteria ▶ Relate Geometric & Topological invariants. (e.g., Volume conjecture ... ) Results (1) Jones polynomial 2-bridge knot Hanselman’s result ▶ ······ Finite type invariants Recent progress ▶ 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 ∪ { 1/0 } 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 ▶ 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 ▶ 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.

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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) ′′ Jones polynomial ̸ 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 ′′ 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 Ozsváthand Szabó, 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 Ozsváth-Szabó. 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)

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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 ′′ ′′′ If a knot K satisﬁes either V (1) ≠ 0 or V (1) ≠ 0 , Results (1) K K Jones polynomial ≇ ′ ′ 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 ′′ ̸ Cosmetic Banding with ∆K (1) = 0 by using the Casson invariant. How To Check? (∆K (t): the normalized Alexander polynomial) ′′ − ′′ 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. Results (1) Jones polynomial 2-bridge knot Hanselman’s result Finite type invariants An: the vector space generated by degree n Jacobi diagrams Recent progress subject to AS and IHX relations Results (2) Genus one alternating Zn: the degree n part of the Kontsevich-Kuperberg-Thurston invariant of knots Cosmetic Banding rational homology spheres taking its value in An How To Check?

Remark: KKT-inv. is a universal ﬁnite type invariant for integral homology spheres (in the sense of Ohtsuki, Habiro and Goussarov)

15 / 36 Cosmetic surgery From λ2 to w3 on knots

K. Ichihara

Introduction 3-manifold Fact [Theorem 7.1, Lescop (2009)] Dehn surgery Cosmetic Surgery Conjecture The invariant λ2 satisfies the surgery formula Known Facts p q ′′ q q Examples 2 Criteria λ2(K( )) = ( ) λ2 (K) + ( ) w3(K) + c( ) a2(K) + λ2(L(p, q)) q p p p Results (1) Jones polynomial ⊂ 3 2-bridge knot for all knots K S . Hanselman’s result Finite type invariants 2 a2(K): the z -coefficient of the Conway polynomial ∇K (z) Recent progress L(p, q): the lens space (obtained by p/q surgery on the unknot) Results (2) ′′ q Genus one alternating λ (K) & c( ): explicit constants defined in [Lescop, 2009] knots 2 p Cosmetic Banding How To Check?

w3(K) is a knot invariant deﬁned by Lescop.

16 / 36 Cosmetic surgery The invariant w3 on knots

K. Ichihara

Introduction Key Lemma 3-manifold Dehn surgery 3 Cosmetic Surgery For all knots K ⊂ S , Conjecture Known Facts 1 ′′′ 1 ′′ Examples w3(K) = VK (1) + VK (1). Criteria 72 24 Results (1) Jones polynomial 2-bridge knot Hanselman’s result This can be shown in the same line as [Prop. 4.2, Nikkuni (2005)] Finite type invariants by using the skein relation for w3 given by Lescop. Recent progress Results (2) Genus one alternating knots Cosmetic Banding Remark: How To Check? ⊂ 3 ′′ − − ′′ For a knot K S , VK (1) = 6a2(K) = 3∆K (1).

17 / 36 Cosmetic surgery vs Heegaard Floer Homology on knots

K. Ichihara

Ozsváthand Szabógave the example of K = 944, which is a genus two knot Introduction 3-manifold such that K(1) and K(−1) have the same Heegaard Floer homology. On the Dehn surgery Cosmetic Surgery other hand, our theorem can detect that they are not homeomorphic. Conjecture Known Facts Examples Corollary Criteria Results (1) The cosmetic surgery conjecture is true for all knots with no more than 11 Jones polynomial 2-bridge knot crossings, except possibly Hanselman’s result Finite type invariants Recent progress 1033, 10118, 10146, 11a91, 11a138, 11a285, 11n86, 11n157. Results (2) Genus one alternating knots Cosmetic Banding Recently, these results are extended by T. Ito by using LMO invariant. How To Check? Remark: LMO-inv. is another universal finite type invariant for ZHS.

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K. Ichihara A knot admitting a diagram with two maxima and minima. Introduction 3-manifold Dehn surgery Cosmetic Surgery Conjecture Known Facts Examples Denote this diagram by C[2, 2, −2, 2, 2, −2] (Conway form). Criteria Results (1) Jones polynomial Also a 2-bridge knot is parametrized by 2-bridge knot Hanselman’s result the slope p/q (irreducible fraction). Finite type invariants Fact Recent progress Results (2) For a 2-bridge knot with S(p, q) & Genus one alternating knots C[a1, . . . , an], Cosmetic Banding How To Check? p/q = [a1, a2, . . . , an] (continued fraction expansion) Denote it by S(p, q) (Schubert form).

20 / 36 Cosmetic surgery Result on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Cosmetic Surgery Theorem [I.-Jong-Mattman-Saito, arXiv:1909.02340] Conjecture Known Facts Two-bridge knots admit no purely cosmetic surgeries. Examples Criteria Results (1) Also, we have the following. Jones polynomial 2-bridge knot Hanselman’s result Finite type invariants Theorem. Recent progress All alternating ﬁbered knots and all alternating pretzel knots admit no purely Results (2) Genus one alternating cosmetic surgeries. knots Cosmetic Banding How To Check?

21 / 36 Cosmetic surgery Hanselman’s result (Heegaard Floer theory) on knots Theorem [Hanselman, arXiv:1906.06773] K. Ichihara 3 ∼ ′ ′ Introduction Let K be a knot in S . Suppose that K(r) = K(r ) for distinct r, r . Then the 3-manifold Dehn surgery following holds. Cosmetic Surgery Conjecture (i) {r, r′} = {2} or {1/q} , ′ Known Facts (ii) if {r, r } = {2}, then g(K) = 2,(g(K)denotes the genus of K) Examples { ′} { } Criteria (iii) if r, r = 1/q , then Results (1) Jones polynomial 2-bridge knot th(K) + 2g(K) Hanselman’s result q ≤ Finite type invariants 2g(K)(g(K) − 1) Recent progress

Results (2) (th(K) denotes the thickness of Heegaard Floer homology) Genus one alternating knots Cosmetic Banding Lemma How To Check? Let K be an alternating knot in S3. If K admits purely cosmetic surgeries, then g(K) = 2 , the signature σ(K) = 0, and the surgery slopes must be 1 or 2.

22 / 36 Cosmetic surgery Jones polynomial on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Cosmetic Surgery By using Jones polynomial of knots, we have the following. Conjecture Known Facts Examples Criteria Lemma ([I.-Wu, CAG, 2019]) Results (1) Jones polynomial If a 2-bridge knot of genus two admitting purely cosmetic surgeries, then it 2-bridge knot Hanselman’s result would be associated to the continued fraction [2x, 2y, −2(x + y), 2x] for integers Finite type invariants x > 0 and y ≠ 0. Recent progress Results (2) Genus one alternating knots Cosmetic Banding How To Check?

23 / 36 Cosmetic surgery Signature σ(K) on knots

K. Ichihara Let K be a 2-bridge knot associated to the continued fraction Introduction [2x, 2y, −2(x + y), 2x] for integers x > 0 and y ≠ 0. 3-manifold Dehn surgery Cosmetic Surgery Proposition 1. Conjecture Known Facts Examples If K admits purely cosmetic surgeries, then y < 0 and (x + y) > 0. Criteria Results (1) We use the following result of [Lee] and [Traczyk]: Jones polynomial 2-bridge knot for a reduced alternating diagram D of an oriented non-split alternating link L, Hanselman’s result Finite type invariants σ(L) = o(D) − y(D) − 1 Recent progress Results (2) Genus one alternating knots Cosmetic Banding It remains to handle the case of y < 0 and (x + y) > 0. How To Check? In this case, the simple (positive) continued fraction for K is [2x − 1, 1, −(2y + 1), 2(x + y) − 1, 1, 2x − 1].

24 / 36 C Cosmetic surgery SL(2, ) Casson invariant on knots

K. Ichihara

Introduction Let K be a 2-bridge knot associated to the continued fraction 3-manifold Dehn surgery − − − − Cosmetic Surgery [2x 1, 1, (2y + 1), 2(x + y) 1, 1, 2x 1] for x > 0, y < 0 with (x + y) > 0. Conjecture Known Facts Examples Proposition 2. Criteria − Results (1) If K admits purely cosmetic surgeries, then x = 2y. Jones polynomial 2-bridge knot Hanselman’s result Note that the knot is amphicheiral when x = −2y. Finite type invariants Recent progress

Results (2) Our key ingredient is the SL(2, C) Casson invariant, originally by [Curtis]. Genus one alternating knots A practical surgery formula for 2-bridge knots was obtained by [Boden-Curtis], Cosmetic Banding How To Check? and was used for a study of cosmetic surgeries on 2-bridge knots in [I.-Saito].

25 / 36 Cosmetic surgery Finite type invariants on knots Note that if x = −2y, then the knot K is associated to the continued fraction K. Ichihara − − [4n, 2n, 2n, 4n] for n > 0. Introduction 3-manifold Dehn surgery Proposition 3. Cosmetic Surgery Conjecture The 2-bridge knot K associated to the continued fraction [4n, −2n, −2n, 4n] for Known Facts Examples a positive integer n admits no purely cosmetic surgeries. Criteria Results (1) Jones polynomial We use the obstructions obtained by [Boyer-Lines] and [Ito]: 2-bridge knot Hanselman’s result If a knot K has a purely cosmetic pair of surgeries, then Finite type invariants ▶ Recent progress a2(K) = 0 ▶ ̸ 4 ̸ 4 Results (2) j4(K) = 14n and j4(K) = 284n for some n > 0. Genus one alternating knots Cosmetic Banding On the other hand, by direct calculations, we have How To Check?

4 4 4 ∇K (z) = 1 + 4n z and j4(K) = −12n

for K = C[4n, −2n, −2n, 4n] with n > 0.

26 / 36 Cosmetic surgery Recent progress on knots

K. Ichihara

[arXiv:2009.00522] Bryan Boehnke, Conan Gillis, Hanwen Liu, Shuhang Xue Introduction The purely cosmetic surgery conjecture is true for the Kinoshita-Terasaka and Conway 3-manifold Dehn surgery knot families Cosmetic Surgery Conjecture [arXiv:2006.06765] AndrásI. Stipsicz, ZoltánSzabó Known Facts Purely cosmetic surgeries and pretzel knots Examples [arXiv:2005.12795] Ina Petkova, Biji Wong Criteria Results (1) Twisted Mazur pattern satellite knots and bordered Floer theory Jones polynomial [arXiv:2005.07278] Konstantinos Varvarezos 2-bridge knot Hanselman’s result 3-braid knots do not admit purely cosmetic surgeries Finite type invariants [arXiv:1909.05048] Ran Tao Recent progress Connected sums of knots do not admit purely cosmetic surgeries Results (2) Genus one alternating [arXiv:1909.02340] Kazuhiro Ichihara, In Dae Jong, Thomas W. Mattman, Toshio Saito knots Cosmetic Banding Two-bridge knots admit no purely cosmetic surgeries (to appear in Algebr.Geom.Topol.) How To Check? [arXiv:1906.06773] Jonathan Hanselman Heegaard Floer homology and cosmetic surgeries in S3 (to appear in J.Eur.Math.Soc.)

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K. Ichihara Recall: A pair of surgeries on K along slopes r1, r2 are chirally cosmetic Introduction if there exists an orientation-reversing. homeo. between K(r1) & K(r2). 3-manifold Dehn surgery Cosmetic Surgery Conjecture amphicheiral knot Known Facts Examples ∀ ̸∈ { } Criteria When K is amphicheiral knot, for slope r 0, 1/0 , Results (1) r-& (−r)-surgeries are chirally cosmetic along equivalent slopes. Jones polynomial 2-bridge knot Hanselman’s result Finite type invariants Recent progress

Mathieu, 1992 Results (2) Genus one alternating knots The (18k + 9)/(3k + 1)- and (18k + 9)/(3k + 2)-surgeries Cosmetic Banding 3 How To Check? on the trefoil knot T2,3 in S are chirally cosmetic for any k ≥ 0.

Remark: The pair of slopes are inequivalent.

29 / 36 Cosmetic surgery Alternating knot of genus one on knots

K. Ichihara Theorem [I.-Ito-Saito] Introduction ′ 3-manifold Let K be an alternating knot of genus one. For distinct slopes r and r , if the r- Dehn surgery ′ Cosmetic Surgery and r -surgeries on K are chirally cosmetic, then either Conjecture (i) K is amphicheiral and r = −r′, or Known Facts { } Examples Criteria { ′} 18k+9 18k+9 (ii) K is the right-handed trefoil and r, r = 3k+1 , 3k+2 , or, K is the Results (1) { } Jones polynomial { ′} − 18k+9 − 18k+9 ∈ Z 2-bridge knot left-handed trefoil and r, r = 3k+1 , 3k+2 (k ). Hanselman’s result Finite type invariants Recent progress

Results (2) Genus one alternating Ingredients: SL(2,C)-Casson inv. & ﬁnite type invariants knots Cosmetic Banding How To Check? Question. Can a non-torus, chiral knot in S3 admit chirally cosmetic surgeries?

30 / 36 Cosmetic surgery New Example on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Theorem [I.-Jong, 2019] Cosmetic Surgery Conjecture There exists a hyperbolic knot with chirally cosmetic surgeries Known Facts Examples along inequivalent slopes yielding hyperbolic manifolds. Criteria Results (1) Jones polynomial 2-bridge knot This gives a counter-example to another conjecture: Hanselman’s result Finite type invariants Conjecture [Bleiler-Hodgson-Weeks], [Kirby’s list Problem 1.81(B)] Recent progress Results (2) Genus one alternating Any hyperbolic knot admits no purely/chirally cosmetic surgeries knots Cosmetic Banding yielding hyperbolic manifolds. How To Check?

31 / 36 Cosmetic surgery Montesinos trick & Banding on knots Montesinos trick [Montesinos, 1975] K. Ichihara ¯ Introduction Let M be a 3-mfd with a double br.cover M along a link L ⊂ M. 3-manifold ¯ ¯ Dehn surgery Suppose that a knot K in M is strongly invertible w.r.t. the axis L. Cosmetic Surgery Conjecture Then, for a slope r ∈ Z, the mfd. K(r) is homeomorphic to Known Facts Examples the double br.cover along the link obtained from L by a banding. Criteria Results (1)

2q ] 4 1 Jones polynomial − ± 2-bridge knot Hanselman’s result ··· Finite type invariants Recent progress

Results (2) Genus one alternating knots q q Cosmetic Banding q How To Check? ··· ··· ] ] ] ···

32 / 36 Kq ± Cosmetic surgery Bleiler-Hodgson-Weeks’s Example on knots

K. Ichihara

Introduction 3-manifold Bleiler-Hodgson-Weeks’s Dehn surgery Cosmetic Surgery example (1999); Conjecture Known Facts yielding L(49, −19). Examples Criteria Results (1) Jones polynomial 2-bridge knot Hanselman’s result ↓ (Cosmetic banding on the knot 9 ) Finite type invariants 27 Recent progress

Results (2) Genus one alternating knots Cosmetic Banding How To Check?

33 / 36 Cosmetic surgery Bleiler-Hodgson-Weeks’s Example on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Cosmetic Surgery mirroring Conjecture Known Facts & 2π/3-rot. Examples Criteria Results (1) Jones polynomial 2-bridge knot Hanselman’s result banding Finite type invariants of slope 0 Recent progress Results (2) Genus one alternating knots Cosmetic Banding How To Check?

33 / 36 Cosmetic surgery Construction on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Cosmetic Surgery Conjecture ▶ This knot K admits Known Facts Examples a (chirally) cosmetic banding. Criteria ▶ Results (1) The double branched cover M Jones polynomial 2-bridge knot along K is hyperbolic. Hanselman’s result ▶ The knot K¯ in M corresponding to Finite type invariants Recent progress

the banding is hyperbolic. Results (2) ▶ Genus one alternating The knot K¯ is not amphicheiral. knots Cosmetic Banding How To Check?

34 / 36 Cosmetic surgery How to check? on knots

K. Ichihara ▶ The double branched cover M along K is hyperbolic. Introduction ▶ The knot K¯ in M corresponding to the banding is hyperbolic. 3-manifold Dehn surgery Cosmetic Surgery checked by HIKMOT Conjecture Known Facts ▶ Examples The knot K¯ is not amphicheiral. Criteria Results (1) checked by HIKMOT + [Dunfield-Hoffman-Licata] Jones polynomial 2-bridge knot Hanselman’s result Finite type invariants ———————— Recent progress [HIKMOT] N. Hoffman, K. Ichihara, M. Kashiwagi, H. Masai, S. Oishi, and A. Results (2) Genus one alternating Takayasu, Verified Computations for Hyperbolic 3-Manifolds, knots Cosmetic Banding Exp. Math. 25 (2016), no. 1, 66–78. How To Check? [Dunfield-Hoffman-Licata] N.M. Dunfield, N.R. Hoffman, and J.E. Licata, Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling, to appear in Math. Res. Letters.

35 / 36 Cosmetic surgery Acknowledgements on knots

K. Ichihara

Introduction 3-manifold Dehn surgery Cosmetic Surgery Conjecture Known Facts Examples Criteria Thank you for your attention! 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?

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