KazuhiroIchihara The SL (2 , C ) Cassoninvariantandcosmeticsurgeries

The SL(2, C ) Casson invariant and cosmetic surgeries

Kazuhiro Ichihara

Nihon University, College of Humanities and Sciences

Joint work with

Toshio Saito

(Joetsu Univ. of Education) Fundamental Groups, Representations and Geometric

Structures in 3-Manifold Topology November 22, 2016 @ Hiroshima University

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SL(2, C ) Casson invariant

Definition. [very rough]

For a closed orientable 3-manifold Σ = W

W

, the

Casson invariant λ

(Σ) is defined as an oriented intersection number of X

(W

) and X

(W

) in X

(F ) which counts only compact, zero-dimensional components of the intersection.

C. L. Curtis, An intersection theory count of the SL

(C)- representations of the fundamental group of a 3-manifold, Topology

(2001), no. 4, 773–787.

H. U. Boden and C. L. Curtis, The SL(2,

) Casson invariant for Dehn surgeries on two-bridge knots, Algebr. Geom. Topol.

12

(2012), no. 4, 2095–2126.

SL(2, C ) Casson invariant

Σ: a closed, orientable 3–manifold (W

, W

, F ): a Heegaard splitting of Σ

Then the inclusions F ,

W

, W

,

Σ induce surjections on π

. X(N ): the character variety for a manifold N

i.e., the set of characters of SL(2,

representations of π

(N ).

Then we have the following diagram:

X(Σ) = X(W

)

X(W

)

X(W

)

↓ ↓

X(W

)

X(F )

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

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SL(2, C ) Casson invariant

X

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

• There exist a compact neighborhoodU of thezero-dimensional components ofX^∗(W1)∩X^∗(W2)⊂X^∗(F)which is disjoint from the higher dimensional components of the intersection, and

• an isotopyh:X^∗(F)→X^∗(F)supported inU such that h(X^∗(W₁))andX^∗(W₂)intersect transversely inU.

Given a zero-dimensional component{χ}ofh(X^∗(W1))∩X^∗(W2), we setεχ=±1, depending on whether the orientation ofh(X^∗(W1)) followed by that ofX^∗(W₂)agrees with the orientation ofX^∗(F)atχ.

Definition. (SL(2, C ) Casson invariant) Define λ

(Σ) =

χ

ε

, where the sum is taken over all the

zero-dimensional components of h(X

(W

))

X

(W

).

Surgery formula

For a knot K in a closed 3–manifold Σ, we denote by

the 3–manifold obtained by Dehn surgery on K along slope p/q.

Surgery formula of λ

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

Then, there exists E

, E

depending only on K such that for every admissible slope p/q, we have

λ

(Σ

(K )) = 1

2

E

.

Here

is the

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.

I

: X(M )

a function for each ξ

H

(∂M ;

) by using H

(∂M ;

= π

(∂M ) and I

(χ) = χ(γ) for γ

π

(M ).

f

: X(M )

the regular function defined by

f

= I

2 for ξ

H

(∂M ;

).

r : X(M )

X(∂M ) the map induced by π

(∂M )

π

M.

Total Culler-Shalen seminorm

Let{Xi} be the collection of all one-dimensional components ofX(M) such thatdimr(Xi) = 1andXi∩X^∗(M)6=∅.

fi,ξ:Xi →C the regular function obtained by restrictingfξ toXi. For the smooth, projective curveXei birationally equivalent toXi, denote the natural extension offi,ξ toXei by f˜i,ξ:Xei→CP¹ . For suchXi, define the seminormk · ki onH1(∂M;R)by setting

kξk_i= deg( ˜f_i,ξ) for allξin the latticeH₁(∂M;Z).

Definition. (the total Culler-Shalen seminorm)

kp/qkCS=X

i

mikp/qki

wheremi>0is the intersection multiplicity ofXi as a curve in the intersectionX^∗(W1)·X^∗(W2)inX(F).

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Admissible slope

Admissible slope

A slope p/q on ∂M is called admissible for a knot K if

1.

p/q is a

slope which is not a strict

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

on ∂M is called regular if there are no irreducible representation ρ : π

(M)

SL(2,

) satisfying that

1.

the character χ

lies on a one-dimensional component X

of X(M ) such that r(X

) is one-dimensional;

2.

trρ(α) =

for all α in the image of i

: π

(∂M )

π

(M );

3.

ker(ρ

i

) is the cyclic group generated by [γ]

π

(∂M ).

Cosmetic surgery conjecture

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

if

orientation preserving homeo. between the manifolds obtained by the surgeries.

Remark: There exists some example of knots admitting

“chirally” cosmetic surgeries along inequivalent slopes.

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Boundary slopes & Cosmetic surgeries Let K be a small knot in an integral homology sphere Σ, and

be the boundary slope set of K (B

Fact.(c.f. Culler-Luecke-Gordon-Shalen)

∃w_j ≥

0 such that

= 2

βj∈B_K

w

∆(γ, β

).

Proposition. (I.-Saito, in progress) If

, then

λ

(Σ

(K)))

λ

(Σ

(K))

for admissible slopes r,

Σ = S

, such a knot

K have no purely cosmetic surgeries along admissible slopes.

2-bridge knots

Proposition. (2-bridge knots with at most 9 crossings) All the two-bridge knots of at most 9 crossings

other than 9

admits no cosmetic surgery pairs.

Remark: 9

= S(49, 19) = C[2, 2,

2, 2,

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Family including 9

Theorem. [I.-Saito] (A family including 9

)

Let K

be a 2-bridge knot C[2x, 2

2x, 2x, 2,

with x

1.

Then K

admits no cosmetic surgeries yielding homology 3-spheres.

i.e., any

- and

-surgeries are not purely cosmetic for K

. Remark:

For K

, the known restrictions cannot be applied.

(original Casson invariant & Heegaard Floer homology)

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

λ

(K(p/q)) =

2kp/qk_CS

if p is even,

1

2kp/qk_CS−

(α

1)/4 if p is odd.

Boden-Curtis (2012)

All slopes are regular for a 2-bridge knot.

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Culler-Shalen norm & Ohtsuki’s method

Boden-Curtis, based on Ohtsuki (1994)

||p/q||_CS

= 1

2

+

i

W

∆(p/q, N

)

!

Here N

,

, N

denotes the boundary slope for K, and W

:=

j

(|n

1) for the continued fraction expansion [n

,

, n

] associated to N

.

T. Ohtsuki, Ideal points and incompressible surfaces in two- bridge knot complements, J. Math. Soc. Japan

(1994), no. 1, 51–87.

Computing Boundary slope

Mattman-Maybrun-Robinson (2008)

The boundary slopes of 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:

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

[b0,2b1+ 1, b2, b3, . . . , bn]7→[b0+ 1,(−2,2)^b1,−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 bound- ary slopes: diameter and genus, Osaka J. Math. 45(2008), 471–489.

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Recent progress

Theorem. [I.-Zhongtao Wu]

Let V

(t) be the Jones polynomial of a knot K in S

. If a knot K satisfies either V

(1)

or V

(1)

,

then K(r) K(r

) as oriented mfds. for distinct slopes r and r

. Lescop’s λ

invariant (2009)

The invariant

:= W

Z

, where W

is a linear form on

with W

( ) = 1 and W

( ) = 0.

A_n

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

Z_n

: the degree n part of the

of rational homology spheres taking its value in