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) f 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 9homeo. of the exterior of K taking one slope to the other. • Two surgeries on K are called purely cosmetic if 9orientation 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. • Two slopes for a knot K are called equivalent if 9homeo. of the exterior of K taking one slope to the other. • Two surgeries on K are called purely cosmetic if 9orientation preserving homeo. between the manifolds obtained by the surgeries. 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 fχg of h(X (W1)) \ X (W2), ∗ we set "χ = 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) P Define λSL(2;C)(Σ) = χ "χ, 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. 8 / 31 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. ∗ ∗ Given a 0-dimensional component fχg of h(X (W1)) \ X (W2), ∗ we set "χ = 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) P Define λSL(2;C)(Σ) = χ "χ, where the sum is taken over all the ∗ ∗ 0-dimensional components of h(X (W1)) \ X (W2). 8 / 31 Cosmetic surgery SL(2; C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress 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. 2 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)) = kp=qk − E : SL(2; ) K 2 CS σ(p) k k Here p=q CS is the total Culler-Shalen semi-norm of the slope p=q and σ(p) ≡ p (mod 2). 9 / 31 Cosmetic surgery SL(2; C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress 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 2 H1(@M) = π1(@M) defined by Iξ(χ) = χ(ξ) for χ 2 X(M). fξ : X(M) ! C the regular function defined by fξ = Iξ − 2 for ξ 2 H1(@M; Z). r : X(M) ! X(@M) the map induced by π1(@M) ! π1M. 10 / 31 Cosmetic surgery SL(2; C) Casson invariant Results (1) Degree 2 part of ZKKT Results (2) Recent progress Total Culler-Shalen seminorm Let fXig be the collection of all one-dimensional components of X(M) ∗ such that dim r(Xi) = 1 and Xi \ X (M) =6 ;. 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 k · ki on H1(@M; R) by setting ~ kξki = deg(fi,ξ) for all ξ in the lattice H1(@M; Z). Definition. (the total Culler-Shalen semi-norm) X kp=qkCS = mikp=qki i where mi > 0 is the intersection multiplicity of Xi as a curve in the ∗ ∗ intersection X (W1) · X (W2) in X(F ).