ToshioSaito (JoetsuUniv.Edu.), ZhongtaoWu (CUHK)Invariantsof3-manifoldsrelatedtotheCassoninvariant2017.1.25-27,RIMS,Kyotouniversity BasedonJointworkswith GeneralizationsoftheCassoninvariantandtheirapplicationstothecosmeticsurgeryconjectureKazuhiroIchihara

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.) ,

(CUHK) Invariants of 3-manifolds related to the Casson invariant

2017.1.25-27, RIMS, Kyoto university

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Papers

•

(with Toshio Saito)

Cosmetic surgery and the SL(2,

) 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

Dehn surgery

Cosmetic surgery conjecture

Casson invariant

Definitions Surgery formula Results (1)

2-bridge knots Outline of Proof Degree 2 part of

Definition Results (2)

Jones polynomial

Corollaries

Recent progress

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Dehn surgery on a knot

K

: a

(i.e., embedded circle) in a 3-manifold

Dehn surgery on K (operation to produce a “NEW” 3-mfd)

1) remove the open neighborhood ofKfromM

(to obtain theexterior E(K)ofK) 2) glue a solid torus back (along a slopeγ)

γ m

f

We denote the obtained manifold by

or, by

if K is a knot in S

.

3 / 31

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Cosmetic surgery conjecture It is natural to ask:

Can a non-trivial Dehn surgery give the same manifold?

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.

Our approach: Using some invariants of 3-manifolds.

4 / 31

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT 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

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.

Our approach: Using some invariants of 3-manifolds.

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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 on the

T

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

Definitions Surgery formula Results (1)

2-bridge knots Outline of Proof Degree 2 part of

Definition Results (2)

Jones polynomial

Corollaries

Recent progress

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.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

SL(2, C ) Casson invariant Settings

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 ) has the structure of complex affine algebraic set.

7 / 31

SL(2, C ) Casson invariant Settings

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

.

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 ) has the structure of complex affine algebraic set.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

SL(2, C ) Casson invariant

X

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

• Consider the0-dimensional components of

X^∗(W1)∩X^∗(W2)⊂X^∗(F), take a compact neighborhoodU which is disjoint from the higher dimensional components, and

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

Given a 0-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 0-dimensional components of h(X

(W

))

X

(W

).

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SL(2, C ) Casson invariant

X

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

• Consider the0-dimensional components of

X^∗(W1)∩X^∗(W2)⊂X^∗(F), take a compact neighborhoodU which is disjoint from the higher dimensional components, and

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

Given a 0-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 0-dimensional components of h(X

(W

))

X

(W

).

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

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 exist E

, E

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

λ

(Σ

(p/q)) = 1

2

E

.

Here

p/q

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 )

the function for ξ

H

(∂M ) = π

(∂M ) defined by I

(χ) = χ(ξ) for χ

X(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.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Total Culler-Shalen seminorm

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

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 semi-norm∥ · ∥i onH1(∂M;R)by setting

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

Definition. (the total Culler-Shalen semi-norm)

∥p/q∥CS=∑

i

mi∥p/q∥i

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) =

2 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 ).

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Table of contents

Cosmetic surgery

Dehn surgery

Cosmetic surgery conjecture

Casson invariant

Definitions Surgery formula

2-bridge knots Outline of Proof Degree 2 part of

Definition Results (2)

Jones polynomial

Corollaries

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 9

admits no cosmetic surgery pairs.

Remark: 9

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

2, 2, 2,

2]

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Let K be a nontrivial knot in S

. Boyer-Lines (1990)

K has no cosmetic surgery pairs if K satisfies

Remark:

∆

(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 r

and r

are purely cosmetic, then r

, r

satisfy that

(a) r

=

r

,

(b) q

1 mod p for r

= p/q,

(c)

(the invariant defined by Ozsv´ ath-Szab´ o).

Remark: They use Heegaard Floer homology.

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Remark:

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

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

Name Schubert Form Alexander Polynomial ∆^′′_K(1)

41 S(5,2) t⁻¹−3 +t 2

61 S(9,7) 2t⁻¹−5 + 2t 4

63 S(13,5) t⁻²−3t⁻¹+ 5−3t+t² 2 77 S(21,8) t⁻²−5t⁻¹+ 9−5t+t² -2

81 S(13,11) 3t⁻¹−7 + 3t 6

83 S(17,4) 4t⁻¹−9 + 4t 8

88 S(25,9) 2t⁻²−6t⁻¹+ 9−6t+ 2t² 4 89 S(25,7) t⁻³−3t⁻²+ 5t⁻¹−7 + 5t−3t²+t³ 4 812 S(29,12) t⁻²−7t⁻¹+ 13−7t+t² -6 813 S(29,11) 2t⁻²−7t⁻¹+ 11−7t+ 2t² 2 914 S(37,14) 2t⁻²−9t⁻¹+ 15−9t+ 2t² -2 919 S(41,16) 2t⁻²−10t⁻¹+ 17−10t+ 2t² -4 927 S(49,19) t⁻³−5t⁻²+ 11t⁻¹−15 + 11t−5t²+t³ 0

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Family including 9

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

)

Let K

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

2x, 2x, 2,

2x] 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)

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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

λ

(K(p/q)) =

2∥

p/q

if p is even,

1

2∥

p/q

(α

1)/4 if p is odd.

Boden-Curtis (2012)

All slopes are regular for a 2-bridge knot.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

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

be the boundary slope set of K (

).

Fact.(c.f. Culler-Gordon-Luecke-Shalen, Boyer-Zhang)

∃

w

0 such that

γ

= 2

Nj∈BK

w

∆(γ, N

).

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)

||

p/q

= 1 2

(

−|

p

+

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

] of α/β associated to N

.

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

(1994), no. 1, 51–87.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

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:

[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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Dehn surgery

Cosmetic surgery conjecture

Casson invariant

Definitions Surgery formula Results (1)

2-bridge knots Outline of Proof

Definition Results (2)

Jones polynomial

Corollaries

Recent progress

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Degree 2 part of Z

Zn

: the degree n part of the

of rational homology spheres taking its value in

An

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

Lescop’s λ

invariant (2009)

The invariant

:= W

Z

, where W

is a linear form on

with W

( ) = 1 and W

( ) = 0.

Remark: the degree 1 part Z

gives the Casson invariant.

21 / 31

Dehn surgery

Cosmetic surgery conjecture

Casson invariant

Definitions Surgery formula Results (1)

2-bridge knots Outline of Proof Degree 2 part of

Definition

Jones polynomial

Corollaries

Recent progress

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Jones polynomial

Our next result gives a severe restriction for a knot in S

to admit purely cosmetic surgery in terms of its Jones polynomial.

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

.

Remark

Boyer and Lines obtained a similar result for a knot K with ∆

(1)

= 0 by using the Casson invariant.

(∆

(t): the normalized Alexander polynomial) Since V

(1) =

3∆

(1), our result can be viewed as

an extension of [Proposition 5.1, Boyer-Lines (1990)].

22 / 31

Jones polynomial

Our next result gives a severe restriction for a knot in S

to admit purely cosmetic surgery in terms of its Jones polynomial.

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

. Remark

Boyer and Lines obtained a similar result for a knot K with ∆

(1)

= 0 by using the Casson invariant.

(∆

(t): the normalized Alexander polynomial) Since V

(1) =

3∆

(1), our result can be viewed as

an extension of [Proposition 5.1, Boyer-Lines (1990)].

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

From λ

to w

Fact [Theorem 7.1, Lescop (2009)]

The invariant λ

satisfies the surgery formula

q)) = (q

p)²λ^′′₂(K) + (q

p)w3(K) +c(q

p)a2(K) +λ2(L(p, q))

for all knots K

S

.

a

(K): the z

-coefficient of the Conway polynomial

(z) L(p, q): the lens space (obtained by p/q surgery on the unknot) λ

(K) & c(

): explicit constants defined in [Lescop, 2009]

w3(K)

is a knot invariant, which is shown as follows.

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The invariant w

Lemma

For all knots K

S

, w

(K) = 1

72 V

(1) + 1

24 V

(1).

This can be shown in the same line as [Prop. 4.2, Nikkuni (2005)]

by using the skein relation for w

given by Lescop together with results in [H.Murakami (1986)].

Remark:

For all knots K

S

, V

(1) =

6a

(K ) =

3∆

(1).

24 / 31

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Outline of Proof

Suppose that a knot K has either V

(1)

= 0 or V

(1)

= 0 .

Case: V

(1)

= 0

Since V

(1) =

3∆

(1), by [Proposition 5.1, Boyer-Lines (1990)], K(r)

K(r

) as oriented mfds. for distinct slopes r and r

. Case: V

(1)

with V

(1) = 0

w

(K)

by Lemma. Now assume that K(r)

= K(r

) for r, r

as oriented mfds. Then, by [Theorem 1.2, Ni-Wu (2015)],

must hold.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Outline of Proof

Suppose that a knot K has either V

(1)

= 0 or V

(1)

= 0 . Case: V

(1)

= 0

Since V

(1) =

3∆

(1), by [Proposition 5.1, Boyer-Lines (1990)], K(r)

K(r

) as oriented mfds. for distinct slopes r and r

.

Now assume that K(r)

= K(r

) for r, r

as oriented mfds. Then, by [Theorem 1.2, Ni-Wu (2015)],

must hold.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Outline of Proof

Suppose that a knot K has either V

(1)

= 0 or V

(1)

= 0 . Case: V

(1)

= 0

Since V

(1) =

3∆

(1), by [Proposition 5.1, Boyer-Lines (1990)], K(r)

K(r

) as oriented mfds. for distinct slopes r and r

. Case: V

(1)

with V

(1) = 0

w

(K)

by Lemma.

Now assume that K(r)

= K(r

) for r, r

as oriented mfds. Then, by [Theorem 1.2, Ni-Wu (2015)],

must hold.

25 / 31

Outline of Proof

Suppose that a knot K has either V

(1)

= 0 or V

(1)

= 0 . Case: V

(1)

= 0

Since V

(1) =

3∆

(1), by [Proposition 5.1, Boyer-Lines (1990)], K(r)

K(r

) as oriented mfds. for distinct slopes r and r

. Case: V

(1)

with V

(1) = 0

w

(K)

by Lemma.

Now assume that K(r)

= K(r

) for r, r

as oriented mfds.

Then, by [Theorem 1.2, Ni-Wu (2015)],

must hold.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Let us consider λ

(K(p/q)) & λ

(K(−p/q)).

Applying the surgery formula for λ

, with V

(1) = a

(K) = 0,

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, q

1 (mod p) . Recall:

L(p, q

)

= L(p, q

) as oriented mfds iff q

q

(mod p).

By w

(K)

= 0, λ

(K(

))

= λ

(K(

)) . A contradiction.

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Let us consider λ

(K(p/q)) & λ

(K(−p/q)).

Applying the surgery formula for λ

, with V

(1) = a

(K) = 0,

[Theorem 1.2(b), Ni-Wu (2015)]

If K(p/q)

= K (

p/q) as oriented mfds, q

1 (mod p) . Recall:

L(p, q

)

= L(p, q

) as oriented mfds iff q

q

(mod p).

By w

(K)

= 0, λ

(K(

))

= λ

(K(

)) . A contradiction.

26 / 31

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Let us consider λ

(K(p/q)) & λ

(K(−p/q)).

Applying the surgery formula for λ

, with V

(1) = a

(K) = 0,

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, q

1 (mod p) . Recall:

L(p, q

)

= L(p, q

) as oriented mfds iff q

q

(mod p).

By w

(K)

= 0, λ

(K(

))

= λ

(K(

)) . A contradiction.

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Finite type invariants of degrees ≤ 3

Corollary

If a knot K has the finite type invariants v

(K)

= 0 or v

(K)

= 0, then K(r)

K(r

) for any two distinct slopes r and r

.

v

& v

: the finite type invariants of order 2 and 3 respectively normalized by the conditions that

·

v

(m(K)) = v

(K) and v

(m(K)) =

v

(K)

for any knot K and its mirror image m(K),

·

v

(3

) = v

(3

) = 1 for the right hand trefoil 3

.

Then we see that: v

(K) = a

(K) =

V

(1) v

(K) =

2w

(K) =

(V

(1) + 3V

(1)).

27 / 31

Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT 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

10

, 10

, 10

,

11a

, 11a

, 11a

, 11n

, 11n

.

Remark

Ozsv´ ath and Szab´ o gave the example of K = 9

, 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

K_{b1,c1,···,bm,cm}: 2-bridge knot of Conway formC(2b1,2c1,· · ·,2bm,2cm)

. Corollary

The knotK_{x,1,−x,x,1,−x} admits no purely cosmetic surgeries forx≥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

)

= 1

2

∑

k=1

c

(

∑k i=1

b

)

∑m i=1

b

(

∑m k=i

c

)

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Cosmetic surgery SL(2,C)Casson invariant Results (1) Degree 2 part ofZ^KKT Results (2) Recent progress

Recent progress

By using

the degree 3 part of the Le-Murakami-Ohtsuki’s invariant Z

, 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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