Generalizations of the Casson Invariant and Their Applications to the Cosmetic Surgery Conjecture Kazuhiro Ichihara

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Generalizations of the Casson invariant and their applications to the cosmetic surgery conjecture

Kazuhiro Ichihara

Nihon University, College of Humanities and Sciences

Based on Joint works with Toshio Saito (Joetsu Univ. Edu.) , Zhongtao Wu (CUHK)

Invariants of 3-manifolds related to the Casson invariant 2017.1.25-27, RIMS, Kyoto university

1 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Papers

• (with Toshio Saito) Cosmetic surgery and the SL(2, C) Casson invariant for two-bridge knots. Preprint, arXiv:1602.02371.

• (with Zhongtao Wu) A note on Jones polynomial and cosmetic surgery. Preprint, arXiv:1606.03372.

2 / 31 Table of contents Cosmetic surgery Dehn surgery Cosmetic surgery conjecture SL(2, C) Casson invariant Deﬁnitions Surgery formula Results (1) 2-bridge knots Outline of Proof Degree 2 part of ZKKT Deﬁnition Results (2) Jones polynomial Corollaries Recent progress Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Dehn surgery on a knot

K : a knot (i.e., embedded circle) in a 3-manifold M

Dehn surgery on K (operation to produce a “NEW” 3-mfd) 1) remove the open neighborhood of K from M (to obtain the exterior E(K) of K) 2) glue a solid torus back (along a slope γ)

γ m

We denote the obtained manifold by MK (γ), or, by K(γ) if K is a knot in S3.

3 / 31 Conjecture. (Problem 1.81(A) in Kirby’s list) Two surgeries on inequivalent slopes are never purely cosmetic.

• Two slopes for a knot K are called equivalent if ∃homeo. of the exterior of K taking one slope to the other. • Two surgeries on K are called purely cosmetic if ∃orientation preserving homeo. between the manifolds obtained by the surgeries.

Our approach: Using some invariants of 3-manifolds.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Cosmetic surgery conjecture

It is natural to ask: Can a non-trivial Dehn surgery give the same manifold?

4 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Cosmetic surgery conjecture

It is natural to ask: Can a non-trivial Dehn surgery give the same manifold?

Conjecture. (Problem 1.81(A) in Kirby’s list) Two surgeries on inequivalent slopes are never purely cosmetic.

Our approach: Using some invariants of 3-manifolds.

4 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Remark

For “Orientation reversing” case, there exist (counter-)examples.

[Mathieu, 1992] There exist some knots admitting “chirally” cosmetic surgeries along inequivalent slopes. Actually (18k + 9)/(3k + 1)- and (18k + 9)/(3k + 2)-surgeries 3 on the trefoil knot T2,3 in S yield orientation-reversingly homeomorphic pairs for any k ≥ 0.

Further examples were obtained by [Rong], [Bleiler-Hodgson-Weeks], [Matignon], [I.-Jong].

5 / 31 Table of contents Cosmetic surgery Dehn surgery Cosmetic surgery conjecture SL(2, C) Casson invariant Deﬁnitions Surgery formula Results (1) 2-bridge knots Outline of Proof Degree 2 part of ZKKT Deﬁnition Results (2) Jones polynomial Corollaries Recent progress Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

SL(2, C) Casson invariant

Deﬁnition. [very rough]

For a closed orientable 3-manifold Σ = W1 ∪F W2, the SL(2, C) Casson invariant λSL(2,C)(Σ) is deﬁned as an oriented intersection ∗ ∗ ∗ number of X (W1) and X (W2) in X (F ) which counts only compact, zero-dimensional components of the intersection.

C. L. Curtis, An intersection theory count of the SL2(C)- representations of the fundamental group of a 3-manifold, Topology 40 (2001), no. 4, 773–787. H. U. Boden and C. L. Curtis, The SL(2, C) Casson invariant for Dehn surgeries on two-bridge knots, Algebr. Geom. Topol. 12 (2012), no. 4, 2095–2126. 6 / 31 X(N): the character variety for a manifold N i.e., the set of characters of SL(2, C) representations of π1(N). Then we have the following diagram:

X(Σ) = X(W1) ∩ X(W2) → X(W1) ↓ ↓

X(W2) → X(F )

NOTE: X(N) has the structure of complex aﬃne algebraic set.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

SL(2, C) Casson invariant

Settings Σ: a closed, orientable 3–manifold (W1,W2,F ): a Heegaard splitting of Σ

Then the inclusions F,→ Wi, Wi ,→ Σ induce surjections on π1.

7 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

SL(2, C) Casson invariant

Settings Σ: a closed, orientable 3–manifold (W1,W2,F ): a Heegaard splitting of Σ

Then the inclusions F,→ Wi, Wi ,→ Σ induce surjections on π1.

X(N): the character variety for a manifold N i.e., the set of characters of SL(2, C) representations of π1(N). Then we have the following diagram:

X(Σ) = X(W1) ∩ X(W2) → X(W1) ↓ ↓

X(W2) → X(F )

NOTE: X(N) has the structure of complex aﬃne algebraic set. 7 / 31 ∗ ∗ Given a 0-dimensional component {χ} of h(X (W1)) ∩ X (W2), ∗ we set εχ = 1, depending on whether the orientation of h(X (W1)) ∗ ∗ followed by that of X (W2) agrees with the orientation of X (F ) at χ.

Deﬁnition. (SL(2, C) Casson invariant) ∑ Deﬁne λSL(2,C)(Σ) = χ εχ, where the sum is taken over all the ∗ ∗ 0-dimensional components of h(X (W1)) ∩ X (W2).

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

SL(2, C) Casson invariant

X∗(Γ): the subspace of characters of irreducible representations.

• Consider the 0-dimensional components of ∗ ∗ ∗ X (W1) ∩ X (W2) ⊂ X (F ), take a compact neighborhood U which is disjoint from the higher dimensional components, and • take an isotopy h : X∗(F ) → X∗(F ) supported in U such that ∗ ∗ h(X (W1)) and X (W2) intersect transversely in U.

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SL(2, C) Casson invariant

X∗(Γ): the subspace of characters of irreducible representations.

∗ ∗ Given a 0-dimensional component {χ} of h(X (W1)) ∩ X (W2), ∗ we set εχ = 1, depending on whether the orientation of h(X (W1)) ∗ ∗ followed by that of X (W2) agrees with the orientation of X (F ) at χ.

Deﬁnition. (SL(2, C) Casson invariant) ∑ Deﬁne λSL(2,C)(Σ) = χ εχ, where the sum is taken over all the ∗ ∗ 0-dimensional components of h(X (W1)) ∩ X (W2).

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Surgery formula

For a knot K in a closed 3–manifold Σ, we denote by ΣK (p/q) the 3–manifold obtained by Dehn surgery on K along slope p/q.

Surgery formula of λSL(2,C) Suppose K is a small knot in an integral homology 3-sphere Σ. ∈ 1 Z Then, there exist E0,E1 2 ≥0 depending only on K such that for every admissible slope p/q, we have 1 λ C (Σ (p/q)) = ∥p/q∥ − E . SL(2, ) K 2 CS σ(p) ∥ ∥ Here p/q CS is the total Culler-Shalen semi-norm of the slope p/q and σ(p) ≡ p (mod 2).

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Total Culler-Shalen seminorm

Suppose K is a small knot in an integral homology 3-sphere Σ with complement M.

Iξ : X(M) → C the function for ξ ∈ H1(∂M) = π1(∂M) deﬁned by Iξ(χ) = χ(ξ) for χ ∈ X(M).

fξ : X(M) → C the regular function deﬁned by fξ = Iξ − 2 for ξ ∈ H1(∂M; Z).

r : X(M) → X(∂M) the map induced by π1(∂M) → π1M.

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Total Culler-Shalen seminorm

Let {Xi} be the collection of all one-dimensional components of X(M) ∗ such that dim r(Xi) = 1 and Xi ∩ X (M) ≠ ∅.

fi,ξ : Xi → C the regular function obtained by restricting fξ to Xi. e For the smooth, projective curve Xi birationally equivalent to Xi, e ˜ e 1 denote the natural extension of fi,ξ to Xi by fi,ξ : Xi → CP .

For such Xi, deﬁne the semi-norm ∥ · ∥i on H1(∂M; R) by setting ˜ ∥ξ∥i = deg(fi,ξ) for all ξ in the lattice H1(∂M; Z). Deﬁnition. (the total Culler-Shalen semi-norm) ∑ ∥p/q∥CS = mi∥p/q∥i i

where mi > 0 is the intersection multiplicity of Xi as a curve in the ∗ ∗ intersection X (W1) · X (W2) in X(F ). 11 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Admissible slope

Admissible slope A slope p/q on ∂M is called admissible for a knot K if 1. p/q is a regular slope which is not a strict boundary slope; 2. No p′-th root of unity is a root of the Alexander polynomial of K , where p′ = p if p is odd and p′ = p/2 if p is even.

Regular slope A slope γ on ∂M is called regular if there are no irreducible representation ρ : π1(M) → SL(2, C) satisfying that

1. the character χρ lies on a one-dimensional component Xi of X(M) such that r(Xi) is one-dimensional; ∗ 2. trρ(α) = 2 for all α in the image of i : π1(∂M) → π1(M); ∗ 3. ker(ρ ◦ i ) is the cyclic group generated by [γ] ∈ π1(∂M).

12 / 31 Table of contents Cosmetic surgery Dehn surgery Cosmetic surgery conjecture SL(2, C) Casson invariant Deﬁnitions Surgery formula Results (1) 2-bridge knots Outline of Proof Degree 2 part of ZKKT Deﬁnition Results (2) Jones polynomial Corollaries Recent progress Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

2-bridge knots

Proposition. (2-bridge knots with at most 9 crossings) All the two-bridge knots of at most 9 crossings other than 927 admits no cosmetic surgery pairs.

Remark: 927 = S(49, 19) = C[2, 2, −2, 2, 2, −2]

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Let K be a nontrivial knot in S3.

Boyer-Lines (1990) ′′ ̸ K has no cosmetic surgery pairs if K satisﬁes ∆K (1) = 0. Remark: ∆K (t) denotes the (symmetrized) Alexander polynomial for K. They use the Casson invariant (original, SU(2)-version).

Ni-Wu (2011)

If the surgeries along slopes r1 and r2 are purely cosmetic, then r1, r2 satisfy that (a) r1 = −r2, 2 (b) q ≡ −1 mod p for r1 = p/q, (c) τ(K) = 0 (the invariant defined by Ozsváth-Szabó).

Remark: They use Heegaard Floer homology. 14 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Remark: For alternating knots, τ(K) = σ(K) (signature of K) holds.

Table: 2-bridge knots of at most 9 crossings with τ = 0

′′ Name Schubert Form Alexander Polynomial ∆K (1) −1 41 S(5, 2) t − 3 + t 2 −1 61 S(9, 7) 2t − 5 + 2t 4 −2 −1 2 63 S(13, 5) t − 3t + 5 − 3t + t 2 −2 −1 2 77 S(21, 8) t − 5t + 9 − 5t + t -2 −1 81 S(13, 11) 3t − 7 + 3t 6 −1 83 S(17, 4) 4t − 9 + 4t 8 −2 −1 2 88 S(25, 9) 2t − 6t + 9 − 6t + 2t 4 −3 −2 −1 2 3 89 S(25, 7) t − 3t + 5t − 7 + 5t − 3t + t 4 −2 −1 2 812 S(29, 12) t − 7t + 13 − 7t + t -6 −2 −1 2 813 S(29, 11) 2t − 7t + 11 − 7t + 2t 2 −2 −1 2 914 S(37, 14) 2t − 9t + 15 − 9t + 2t -2 −2 −1 2 919 S(41, 16) 2t − 10t + 17 − 10t + 2t -4 −3 −2 −1 2 3 927 S(49, 19) t − 5t + 11t − 15 + 11t − 5t + t 0

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Family including 927

Theorem. [I.-Saito] (A family including 927)

Let Kx be a 2-bridge knot C[2x, 2 − 2x, 2x, 2, −2x] with x ≥ 1. Then Kx admits no cosmetic surgeries yielding homology 3-spheres.

1 1 i.e., any n - and m -surgeries are not purely cosmetic for Kx.

Remark: For Kx, the known restrictions cannot be applied. (original Casson invariant & Heegaard Floer homology)

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Surgery formula for 2-bridge knots

Boden-Curtis (2012) Let K = S(α, β) be a 2-bridge knot and K(p/q) the 3-manifold obtained by p/q-surgery on K. Suppose that p/q is admissible. Then { 1 ∥p/q∥ if p is even, λ (K(p/q)) = 2 CS SL(2,C) 1 ∥ ∥ − − 2 p/q CS (α 1)/4 if p is odd.

Boden-Curtis (2012) All slopes are regular for a 2-bridge knot.

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Boundary slopes & Cosmetic surgeries

Let K be a small knot in an integral homology sphere Σ, and BK be the boundary slope set of K (BK ⊂ Q). Fact.(c.f. Culler-Gordon-Luecke-Shalen, Boyer-Zhang) ∑ ∃wj ≥ 0 such that ||γ||CS = 2 wj∆(γ, Nj).

Nj ∈BK

Boundary slope A slope r on the toral boundary ∂M of a 3-manifold M is called a boundary slope if r is represented by boundary components of an incompressible and not boundary parallel surface properly embedded in M.

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Culler-Shalen norm & Ohtsuki’s method

Let K = S(α, β) be a 2-bridge knot.

Boden-Curtis, based on Ohtsuki (1994) ( ) 1 ∑ ||p/q|| = −|p| + W ∆(p/q, N ) CS 2 i i i ··· Here N∏1, ,Nn denotes the boundary slope for K, and | | − Wi := j( nj 1) for the continued fraction expansion [n1, ··· , nm] of α/β associated to Ni. T. Ohtsuki, Ideal points and incompressible surfaces in two- bridge knot complements, J. Math. Soc. Japan 46 (1994), no. 1, 51–87.

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Computing Boundary slope

Mattman-Maybrun-Robinson (2008) The boundary slopes of K = S(α, β) are associated to the continued fractions obtained by applying the substitutions at non-adjacent positions in the simple continued fraction of α/β. Substitution 1: b1−1 [b0, 2b1, b2, b3, . . . , bn] 7→ [b0 + 1, (−2, 2) , −2, b2 + 1, b3, . . . , bn] Substitution 2:

b1 [b0, 2b1 + 1, b2, b3, . . . , bn] 7→ [b0 + 1, (−2, 2) , −b2 − 1, −b3,..., −bn]

Recall: The simple continued fraction is the unique one with all terms positive and greater than 1. T. W. Mattman, G. Maybrun and K. Robinson, 2-bridge knot boundary slopes: diameter and genus, Osaka J. Math. 45 (2008), 471–489. 20 / 31 Table of contents Cosmetic surgery Dehn surgery Cosmetic surgery conjecture SL(2, C) Casson invariant Deﬁnitions Surgery formula Results (1) 2-bridge knots Outline of Proof Degree 2 part of ZKKT Deﬁnition Results (2) Jones polynomial Corollaries Recent progress Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Degree 2 part of ZKKT

Zn: the degree n part of the Kontsevich-Kuperberg-Thurston invariant of rational homology spheres taking its value in An

An: the vector space generated by Jacobi diagrams of degree n subject to AS and IHX relations

Lescop’s λ2 invariant (2009)

The invariant λ2 := W2 ◦ Z2, where W2 is a linear form on An

with W2( ) = 1 and W2( ) = 0.

Remark: the degree 1 part Z1 gives the Casson invariant.

21 / 31 Table of contents Cosmetic surgery Dehn surgery Cosmetic surgery conjecture SL(2, C) Casson invariant Deﬁnitions Surgery formula Results (1) 2-bridge knots Outline of Proof Degree 2 part of ZKKT Deﬁnition Results (2) Jones polynomial Corollaries Recent progress Remark Boyer and Lines obtained a similar result for a knot K ′′ ̸ with ∆K (1) = 0 by using the Casson invariant. (∆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)].

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Jones polynomial

Our next result gives a severe restriction for a knot in S3 to admit purely cosmetic surgery in terms of its Jones polynomial.

Theorem [I.-Zhongtao Wu] 3 Let VK (t) be the Jones polynomial of a knot K in S . ′′ ̸ ′′′ ̸ If a knot K satisﬁes either VK (1) = 0 or VK (1) = 0 , then K(r) ≇ K(r′) as oriented mfds. for distinct slopes r and r′.

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Jones polynomial

Our next result gives a severe restriction for a knot in S3 to admit purely cosmetic surgery in terms of its Jones polynomial.

Remark Boyer and Lines obtained a similar result for a knot K ′′ ̸ with ∆K (1) = 0 by using the Casson invariant. (∆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)].

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From λ2 to w3

Fact [Theorem 7.1, Lescop (2009)]

The invariant λ2 satisﬁes the surgery formula p q q q λ (K( )) = ( )2 λ′′(K) + ( ) w (K) + c( ) a (K) + λ (L(p, q)) 2 q p 2 p 3 p 2 2

for all knots K ⊂ S3.

2 a2(K): the z -coeﬃcient of the Conway polynomial ∇K (z) L(p, q): the lens space (obtained by p/q surgery on the unknot) ′′ q λ2(K) & c( p ): explicit constants deﬁned in [Lescop, 2009]

w3(K) is a knot invariant, which is shown as follows.

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The invariant w3

Lemma For all knots K ⊂ S3, 1 1 w (K) = V ′′′(1) + V ′′ (1). 3 72 K 24 K

This can be shown in the same line as [Prop. 4.2, Nikkuni (2005)] by using the skein relation for w3 given by Lescop together with results in [H.Murakami (1986)].

Remark: ⊂ 3 ′′ − − ′′ For all knots K S , VK (1) = 6a2(K) = 3∆K (1).

24 / 31 ′′ ̸ Case: VK (1) = 0 ′′ − ′′ Since VK (1) = 3∆K (1), by [Proposition 5.1, Boyer-Lines (1990)], K(r) ≇ K(r′) as oriented mfds. for distinct slopes r and r′.

′′′ ̸ ′′ ⇒ ̸ Case: VK (1) = 0 with VK (1) = 0 w3(K) = 0 by Lemma.

∼ ′ ′ Now assume that K(r) = K(r ) for r, r ∈ Q as oriented mfds.

Then, by [Theorem 1.2, Ni-Wu (2015)], r = −r′ must hold.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Outline of Proof

′′ ̸ ′′′ ̸ Suppose that a knot K has either VK (1) = 0 or VK (1) = 0 .

25 / 31 ′′′ ̸ ′′ ⇒ ̸ Case: VK (1) = 0 with VK (1) = 0 w3(K) = 0 by Lemma.

∼ ′ ′ Now assume that K(r) = K(r ) for r, r ∈ Q as oriented mfds.

Then, by [Theorem 1.2, Ni-Wu (2015)], r = −r′ must hold.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Outline of Proof

′′ ̸ ′′′ ̸ Suppose that a knot K has either VK (1) = 0 or VK (1) = 0 .

′′ ̸ Case: VK (1) = 0 ′′ − ′′ Since VK (1) = 3∆K (1), by [Proposition 5.1, Boyer-Lines (1990)], K(r) ≇ K(r′) as oriented mfds. for distinct slopes r and r′.

25 / 31 ∼ ′ ′ Now assume that K(r) = K(r ) for r, r ∈ Q as oriented mfds.

Then, by [Theorem 1.2, Ni-Wu (2015)], r = −r′ must hold.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Outline of Proof

′′ ̸ ′′′ ̸ Suppose that a knot K has either VK (1) = 0 or VK (1) = 0 .

′′ ̸ Case: VK (1) = 0 ′′ − ′′ Since VK (1) = 3∆K (1), by [Proposition 5.1, Boyer-Lines (1990)], K(r) ≇ K(r′) as oriented mfds. for distinct slopes r and r′.

′′′ ̸ ′′ ⇒ ̸ Case: VK (1) = 0 with VK (1) = 0 w3(K) = 0 by Lemma.

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Outline of Proof

′′ ̸ ′′′ ̸ Suppose that a knot K has either VK (1) = 0 or VK (1) = 0 .

′′ ̸ Case: VK (1) = 0 ′′ − ′′ Since VK (1) = 3∆K (1), by [Proposition 5.1, Boyer-Lines (1990)], K(r) ≇ K(r′) as oriented mfds. for distinct slopes r and r′.

′′′ ̸ ′′ ⇒ ̸ Case: VK (1) = 0 with VK (1) = 0 w3(K) = 0 by Lemma.

∼ ′ ′ Now assume that K(r) = K(r ) for r, r ∈ Q as oriented mfds.

Then, by [Theorem 1.2, Ni-Wu (2015)], r = −r′ must hold.

25 / 31 ′′ Applying the surgery formula for λ2, with VK (1) = a2(K) = 0,

p − − p q − − λ2(K( q )) λ2(K( q )) = 2( p ) w3(K) + λ2(L(p, q)) λ2(L(p, q))

∼ Here we see that L(p, q) = L(p, −q) as oriented manifolds by;

[Theorem 1.2(b), Ni-Wu (2015)] ∼ If K(p/q) = K(−p/q) as oriented mfds, q2 ≡ −1 (mod p) .

Recall: ∼ ≡ 1 L(p, q1) = L(p, q2) as oriented mfds iﬀ q1 q2 (mod p). ̸ p ̸ − p □ By w3(K) = 0, λ2(K( q )) = λ2(K( q )) . A contradiction.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Let us consider λ2(K(p/q)) & λ2(K(−p/q)).

26 / 31 ∼ Here we see that L(p, q) = L(p, −q) as oriented manifolds by;

[Theorem 1.2(b), Ni-Wu (2015)] ∼ If K(p/q) = K(−p/q) as oriented mfds, q2 ≡ −1 (mod p) .

Recall: ∼ ≡ 1 L(p, q1) = L(p, q2) as oriented mfds iﬀ q1 q2 (mod p). ̸ p ̸ − p □ By w3(K) = 0, λ2(K( q )) = λ2(K( q )) . A contradiction.

Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Let us consider λ2(K(p/q)) & λ2(K(−p/q)). ′′ Applying the surgery formula for λ2, with VK (1) = a2(K) = 0,

p − − p q − − λ2(K( q )) λ2(K( q )) = 2( p ) w3(K) + λ2(L(p, q)) λ2(L(p, q))

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Let us consider λ2(K(p/q)) & λ2(K(−p/q)). ′′ Applying the surgery formula for λ2, with VK (1) = a2(K) = 0,

p − − p q − − λ2(K( q )) λ2(K( q )) = 2( p ) w3(K) + λ2(L(p, q)) λ2(L(p, q))

∼ Here we see that L(p, q) = L(p, −q) as oriented manifolds by;

[Theorem 1.2(b), Ni-Wu (2015)] ∼ If K(p/q) = K(−p/q) as oriented mfds, q2 ≡ −1 (mod p) .

Recall: ∼ ≡ 1 L(p, q1) = L(p, q2) as oriented mfds iﬀ q1 q2 (mod p). ̸ p ̸ − p □ By w3(K) = 0, λ2(K( q )) = λ2(K( q )) . A contradiction.

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Finite type invariants of degrees ≤ 3

Corollary

If a knot K has the ﬁnite type invariants v2(K) ≠ 0 or v3(K) ≠ 0, then K(r) ≇ K(r′) for any two distinct slopes r and r′.

v2 & v3: the ﬁnite type invariants of order 2 and 3 respectively normalized by the conditions that · v2(m(K)) = v2(K) and v3(m(K)) = −v3(K) for any knot K and its mirror image m(K), · v2(31) = v3(31) = 1 for the right hand trefoil 31. − 1 ′′ Then we see that: v2(K) = a2(K) = 6 VK (1) − − 1 ′′′ ′′ v3(K) = 2w3(K) = 36 (VK (1) + 3VK (1)). 27 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

vs Heegaard Floer Homology

Corollary The cosmetic surgery conjecture is true for all knots with no more than 11 crossings, except possibly

1033, 10118, 10146,

11a91, 11a138, 11a285, 11n86, 11n157.

Remark

Ozsv´athand Szab´ogave the example of K = 944, which is a genus two knot such that K(1) and K(−1) have the same Heegaard Floer homology.

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2-bridge knots

··· Kb1,c1,··· ,bm,cm : 2-bridge knot of Conway form C(2b1, 2c1, , 2bm, 2cm). Corollary

The knot Kx,1,−x,x,1,−x admits no purely cosmetic surgeries for x ≥ 1.

Remark: This gives an extension of Theorem [I.-Saito].

To show that, we have the next, which is of interest independently. Proposition

v (K ··· ) = −2w (K ··· ) 3 b1,c1, ,bm,cm ( 3 b1,c1, ,bm,cm ) 1 ∑m ∑k ∑m ∑m = c ( b )2 − b ( c )2 2 k i i k k=1 i=1 i=1 k=i

29 / 31 Cosmetic surgery SL(2, C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress

Recent progress

By using the degree 3 part of the Le-Murakami-Ohtsuki’s invariant ZLMO, Tetsuya Ito obtained the following (private communications):

Theorem (T. Ito) The cosmetic surgery conjecture is true for all knots with less than or equal to 11 crossings.

He also obtained several constrains for existence of cosmetic surgeries for knots by using the LMO-invariant.

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Acknowledgements

Thank you for your attention.

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