H(2)-unknotting operation and Heegaard Floer homology

H(2)-unknotting operation and Heegaard Floer homology

Yuanyuan Bao

Tokyo Institute of Technology

December 21st 2010 Nihon University

H(2)-unknotting operation

Fact: H(2)-move is an unknotting operation.

Definition (Hoste-Nakanishi-Taniyama 1990)

K ⊂S³: a knot. The H(2)-unknotting number of K is:

u₂(K):= the minimal number of H(2)-moves needed to change K into the unknot.

Example

H(2)-move

≈

3 :

Therefore u2(31) = 1.

H(2)-move

4 :

Question(Riley): u2(41) = 1?

Answer(Lickorish,1986): No. u₂(4₁) = 2.

Lickorish’s theorem

Theorem (Lickorish,1986) K ⊂S³.

Σ(K): double branched cover of S³ along K .

If u₂(K) = 1, then H₁(Σ(K),Z)is cyclic of order |det(K)|, and ∃a generator g s.t. the linking form has the following form:

λ:H₁(Σ(K),Z)×H₁(Σ(K),Z) → Q/Z

(g,g) 7→ ±1/det(K).

Example

Figure eight knot 4₁: det(4₁)=5;

H₁(Σ(4₁),Z) =Z/5Z.

The linking form of 4₁ is:

λ:H₁(Σ(4₁),Z)×H₁(Σ(4₁),Z) → Q/Z (a,a) 7→ 2/5.

∃ NO generatorg such that (g,g)7→ ±1/5.

⇒u₂(4₁)>1.

Other criterions

Bleiler (1990): If the H(2)-unknotting number of a composite knot is one, then one summand is a 2-bridge knot.

Taniyama-Yasuhara (1994): Equivalence of H(2)-unknotting number and two invariants from g-concordance.

Yasuhara (1996): Criterion using signature and Arf invariant.

Kanenobu-Miyazawa (2009): Criterians from polynomial invariants.

· · · ·

Main Result

Y: rational homology 3-sphere.

Spin^c(Y): the set of spin^c-structures over Y (=H1(Y,Z) as sets).

If |H₁(Y,Z)|is odd,Spin^c(Y) has a group structure:

Spin^c(Y) → H1(Y,Z) s 7→ [c₁(s)]/2.

s₁+s₂:=s if [c1(s1) +c1(s2)]

2 = [c1(s)]

2 . Herec1(s) is the first chern class associated tos.

From Heegaard Floer homology theory, we have the following map:

d :Spin^c(Y) → Q s 7→ d(Y,s).

Main Result

Theorem (B)

K ⊂S³ with|det(K)|=p.

If u₂(K) = 1, then∃ an isomorphism φ:Z/pZ→Spin^c(Σ(K))and a sign =±1such that for all i ∈Z/pZ

I_φ,(i) :=·d(Σ(K), φ(i))− 1 4(1

p(p+ (−1)ⁱp

2 −1)²−1) = 0 (mod 2), and I_φ,(i) ≤ 0.

Example

P(13,4,11)

Therefore u₂(P(13,4,11))≤2.

Using our method, we can prove u₂(P(13,4,11))>1, and thus u2(P(13,4,11)) = 2.

Some other known criterions including Lickorish’s theorem fail here to detect whetheru₂(P(13,4,11))>1 or not.

Proof-step 1

Lemma(Montesinos’s trick): If u₂(K) = 1, then Σ(K) =±S_p³(C) for some knot C ⊂S³.

The neighborhood of C

Proof-step2

Therefore ±Σ(K) =S_p³(C) bounds a positive-definite 4-mainfoldW.

W =B⁴ [

alongCwith framingp

2−handle.

The intersection form ofW is represented by the 1×1 matrix (p).

Proof-step3

Theorem (Ozsv´ath-Szab´o,2003)

Let Y be a rational homology three-sphere and fix a Spin^c-structure s over Y . Then for each smooth, positive-definite four-manifold X whose

boundary is Y , and for each Spin^c-structure t∈Spin^c(X)with t|Y =s, we have that

c1(t)²−rk(H²(X,Z))

4 ≥d(Y,s).

The definition of d(Y,s) guarantees that c₁(t)²−rk(H²(X,Z))

=d(Y,s) (mod 2).

Proof-step4

The set of characteristic covectors Char((p)) :=

ξ ∈Z

ξ.v =p.v² (mod 2) for anyv ∈Z

= {ξ∈Z|ξ=p (mod 2)}.

Defineξ ∼ζ ⇔(ξ−ζ)/p ∈Z. Then ^Char((p))_∼ =Z/pZ. For the set {c₁(t)|t ∈Spin^c(W)}, define

c₁(t₁)∼c₁(t₂)⇔t₁|S_p³(C) =t₂|S_p³(C).

Then ^{{c1(t)|t∈Spin}_∼ ^c(W)} ∼=Spin^c(S_p³(C)).

Proof-step4

Char((p)) = {c₁(t)|t∈Spin^c(W)}

↓ ↓

Char((p))

∼ → ^{{c1(t)|t∈Spin}_∼ ^c(W)}

k k

Z/pZ → Spin^c(S_p³(C))

&φ k Spin^c(Σ(K))

Char((p)) = {c₁(t)|t∈Spin^c(W)}

Proof-step5

c₁(t)²−rk(H²(W,Z))

4 ≥d(S_p³(C),s) =±d(Σ(K),s)

⇔ ^ξ2/p−1₄ ≥ ±d(Σ(K),s) (∀ξ∈Char((p)) with [ξ] =φ⁻¹(s))

⇔ maxn_ξ2/p−1 4

ξ∈Char((p)),[ξ] =φ⁻¹(s)o

≥ ±d(Σ(K),s)

⇔ maxn

ξ²/p−1

4 |ξ ∈Char((p)),[ξ] =i ∈Z/pZo

≥ ±d(Σ(K), φ(i))

⇔ ¹₄(_p¹(^p+(−1)₂ ^ip −1)²−1)≥ ±d(Σ(K), φ(i)).

Remark1

Theorem(Ozsv´ath-Szab´o,2004): The lens spaces L(α, β) is obtained as a surgery on a knot K ⊂S³ only if there is a one-to-one correspondence

σ:Z/αZ→Spin^c(L(α, β)) with the following symmetries:

σ(−i) =σ(i)

there is an isomorphism ϕ:Z/αZ→Z/αZwith the property that σ(i)−σ(j) =ϕ(i−j),

with the following properties. Fori ∈Z, there is

Remark2

Ozsv´ath and Szab´o (2005) first used this machinery to study knots with unknotting number one.

Owens (2005) extended Ozsv´ath and Szab´o’s idea and studied knots with higher unknotting numbers.

Gilmer and Livingston (2010) applied Ozsv´ath and Szab´o’s theorem to study the nonorientable four-genus of a knot, which is an invariant closed related to H(2)-unknotting number.

Further study

Consider the spectral sequence from the Khovanov homology of a knot to the Heggaard Floer homology of the double branched cover.

If a knot has H(2)-unknotting number one, start from Heegaard Floer homology and reverse the sequence to look for restrictions to Khovanov homology.