Jones polynomial and cosmetic surgery

Introduction Main Theorem Proof Corollaries

Jones polynomial and cosmetic surgery

Kazuhiro Ichihara

(Nihon Univ., College of Humanities & Sciences)

Joint work with

Zhongtao Wu

(Chinese University of Hong Kong) KKOOK 2016

Osaka Electro-Comm. Univ., Aug. 22, 2016.

Introduction Main Theorem Proof Corollaries

Dehn surgery

E(K): the exterior of a

knot K in the 3-sphere S

(i.e., S

−(open tubular neighborhood of K))

Dehn surgery:

Gluing a solid torus to E(K)

along a slope γ ;

f

We denote the obtained mfd. by

when [γ] = r ∈

. Recall:

For a knot K in S

, by using a

system, { slope on ∂N (K) } ←→

p q

∪

1

0

Introduction Main Theorem Proof Corollaries

Cosmetic surgery

Cosmetic Surgery Conjecture [Kirby’s list Problem 1.81(A)]

Dehn surgeries on inequivalent slopes are never purely cosmetic.

•

Two slopes are equivalent if there exists

a homeo. of E(K ) taking one slope to the other.

•

Two surgeries on K along slopes r

, r

are purely cosmetic if there exists an

homeo. between K(r

) & K(r

).

Remark: (Mathieu, 1992)

There exist examples for “orientation reversing” case.

In fact, (18k + 9)/(3k + 1)- and (18k + 9)/(3k + 2)-surgeries

on the trefoil yield ori.-reversingly homeomorphic pairs for k ≥ 0.

Introduction Main Theorem Proof Corollaries

Cosmetic surgery

Cosmetic Surgery Conjecture [Kirby’s list Problem 1.81(A)]

Dehn surgeries on inequivalent slopes are never purely cosmetic.

•

Two slopes are equivalent if there exists

a homeo. of E(K ) taking one slope to the other.

•

Two surgeries on K along slopes r

, r

are purely cosmetic if there exists an

homeo. between K(r

) & K(r

).

Remark: (Mathieu, 1992)

There exist examples for “orientation reversing” case.

In fact, (18k + 9)/(3k + 1)- and (18k + 9)/(3k + 2)-surgeries

on the trefoil yield ori.-reversingly homeomorphic pairs for k ≥ 0.

Introduction Main Theorem Proof Corollaries

Main Theorem

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

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

Main Theorem

Let V

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

. If a knot K satisfies either V

(1) 6= 0 or V

(1) 6= 0 ,

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) 6= 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)].

Introduction Main Theorem Proof Corollaries

Main Theorem

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

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

Main Theorem

Let V

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

. If a knot K satisfies either V

(1) 6= 0 or V

(1) 6= 0 ,

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) 6= 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)].

Introduction Main Theorem Proof Corollaries

Lescop’s λ

invariant

The essential new ingredient for our result is the following.

Lescop’s λ

invariant (2009)

The invariant

:= W

◦ Z

, where W

is a linear form on A

with W

( ) = 1 and W

( ) = 0.

A

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

Zn

: the degree n part of the

of rational homology spheres taking its value in A

Introduction Main Theorem Proof Corollaries

From λ

to w

Fact [Theorem 7.1, Lescop (2009)]

The invariant λ

satisfies the surgery formula

q)) = (q

p)²λ⁰⁰₂(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.

Introduction Main Theorem Proof Corollaries

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.

Remark:

For all knots K ⊂ S

, V

(1) = −6a

(K ) = −3∆

(1).

Introduction Main Theorem Proof Corollaries

Outline of Proof of Main Theorem

Suppose that a knot K has either V

(1) 6= 0 or V

(1) 6= 0 .

Case: V

(1) 6= 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) 6= 0 with V

(1) = 0 ⇒ w

(K) 6= 0 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.

Introduction Main Theorem Proof Corollaries

Outline of Proof of Main Theorem

Suppose that a knot K has either V

(1) 6= 0 or V

(1) 6= 0 . Case: V

(1) 6= 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) 6= 0 with V

(1) = 0 ⇒ w

(K) 6= 0 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.

Introduction Main Theorem Proof Corollaries

Outline of Proof of Main Theorem

Suppose that a knot K has either V

(1) 6= 0 or V

(1) 6= 0 . Case: V

(1) 6= 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) 6= 0 with V

(1) = 0 ⇒ w

(K) 6= 0 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.

Introduction Main Theorem Proof Corollaries

Outline of Proof of Main Theorem

Suppose that a knot K has either V

(1) 6= 0 or V

(1) 6= 0 . Case: V

(1) 6= 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) 6= 0 with V

(1) = 0 ⇒ w

(K) 6= 0 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.

Introduction Main Theorem Proof Corollaries

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) 6= 0, λ

(K(

)) 6= λ

(K(−

)) . A contradiction.

Introduction Main Theorem Proof Corollaries

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) 6= 0, λ

(K(

)) 6= λ

(K(−

)) . A contradiction.

Introduction Main Theorem Proof Corollaries

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) 6= 0, λ

(K(

)) 6= λ

(K(−

)) . A contradiction.

Introduction Main Theorem Proof Corollaries

Finite type invariants of degrees ≤ 3

Corollary

If a knot K has the finite type invariants v

(K) 6= 0 or v

(K) 6= 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)).

Introduction Main Theorem Proof Corollaries

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.

Introduction Main Theorem Proof Corollaries

2-bridge knots

Kb₁,c₁,···,bm,c_m: 2-bridge knot of Conway formC(2b1,2c1,· · ·,2bm,2cm)

. Corollary

The knotKx,1,−x,x,1,−x admits no purely cosmetic surgeries forx≥1.

Remark: Known methods cannot obtain this (cf. I-Saito, 2016).

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

Proposition

v

(K

) = −2w

(K

)

= 1

2

m

X

k=1

c

(

k

X

i=1

b

)

−

m

X

i=1

b

(

m

X

k=i

c

)

!

Introduction Main Theorem Proof Corollaries

Whitehead double

D

(K, n): the

n-twisted Whitehead double of a knot K Corollary

There is no purely cosmetic surgery for the positive n-twisted Whitehead double D

(K, n) for n 6= 0.

Moreover, if a

(K ) 6= 0, then there is no purely cosmetic surgery

for the untwisted Whitehead double D

(K, 0) .