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Categorified Vassiliev skein relation on Khovanov homology Jun Yoshida (joint work with Noboru Ito) The University of Tokyo, Graduate School of Mathemtaical Sciences December 19, 2019 Introduction Review on Khovanov homology Vassiliev skein

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Categorified Vassiliev skein relation on Khovanov homology

Jun Yoshida (joint work with Noboru Ito)

The University of Tokyo, Graduate School of Mathemtaical Sciences

December 19, 2019

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Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference

Contents

1

Introduction Goal of the talk Background: Vassiliev invariants Backgound: categorification Motivating Question Goal of the talk (continued) Main Theorem 1 Main Theorem 2

2

Review on Khovanov homology As a bigraded module On differential

3

Vassiliev skein relation on Khovanov homology Mapping cone as subtraction Genus-1 map Invariance under elementary moves Cube of resolutions Example

4

Main Results Khovanov homology on singular links Example: “figure-eight” FI-relation

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Introduction

1

Introduction Goal of the talk Background: Vassiliev invariants Backgound: categorification Motivating Question Goal of the talk (continued) Main Theorem 1 Main Theorem 2

2

Review on Khovanov homology

3

Vassiliev skein relation on Khovanov homology

4

Main Results

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Goal of the talk

Goal To investigate Vassiliev skein relation on Khovanov homology. Recall v: a polynomial link invariant. Vassiliev skein relation: v

  • = v
  • − v
  • .

Extension of v to singular knots/links. Main Question What is the appropriate notion of “crossing change” on Khovanov homology? What properties does extended Khovanov homology enjoy? e.g. FI-relation, 4T-relation, etc...

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Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference

Background: Vassiliev invariants

Convention Singular knot/link ≡ immersed closed 1-manifold in S3 whose singular values are at worst finite and all transverse double points. Ambient isotopy classes are mainly considered. Definition A polynomial invariant v is said to be of type n if v(K) = 0 for every knot K with at least n double points. v is called a Vassiliev invariant if it is of type n for some n. Quantum invariants yield Vassiliev invariants [Birman and Lin, 1993] Jones polynomial ∈ Quantum Invariants Vassiliev Invariants Taylor expansion

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Backgound: categorification

Khovanov homology Khi,j(L) Link invariant values in bigraded abelian groups. Theorem ([Khovanov, 2000]) For a link L, the graded Euler characteristic of Kh(L), i.e. [Kh(L)]q :=

  • i,j

(−1)iqj dim Khi,j(L) ∈ Z[q, q−1] , equals the unnormalized Jones polynomial of L. Slogan: Khovanov homology categorifies Jones polynomial: if V (L) is the Jones polynomial of a link L, then [Kh(L)]q = (q + q−1)V (L) .

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Motivating Question

Kh V Vassiliev Invariants

???

Taylor expansion decategorify categorify ??? ??? Question

1 How does Khovanov homology relate to Vassiliev invariants? 2 Furthermore, does it produces categorifications of Vassiliev

invariants?

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Goal of the talk (continued)

Target Question What is the counterpart of Vassiliev skein relation on Khovanov homology? Approach Consider long exact sequence instead of subtraction. Recall that a long exact sequence of cohomologies · · · → Hn(X) → Hn(Y ) → Hn(Z) → Hn+1(X) → . . . yields the identity on the Euler characteristics χ: χ(X) − χ(Y ) + χ(Z) = 0 . Compare it with Vassiliev skein relation: v

  • − v
  • + v
  • = 0

.

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Main Theorem 1

Theorem (Ito-Y. arXiv:1911.09308) Khovanov homology (with coefficients in F2) extends to a singular link invariant so that there is a long exact sequence · · · → Khi,j

  • ; F2
  • Φ∗

− − → Khi,j

  • ; F2
  • → Khi,j
  • ; F2
  • → Khi+1,j
  • ; F2
  • → . . .

. Remark The invariant seems to be NEW! The long exact sequence can be seen as a categorified Vassiliev skein relation. The map Φ has an easy description.

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Main Theorem 1

Corollary Khovanov homology categorifes unnormalized Jones polynomial even

  • n singular links.

Proof Observe that the identity

  • Kh
  • ; F2
  • q

−

  • Kh
  • ; F2
  • q

+

  • Kh
  • ; F2
  • q

= 0 arising from the long exact sequence is exactly the Vassiliev skein relation for unnormalized Jones polynomial.

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Main Theorem 2

Theorem (Ito-Y. arXiv:1911.09308) Khovanov homology (with coefficients in F2) satisfies FI-relation; i.e. Kh

  • ; F2
  • = 0

. Equivalently The genus-1 map Φ is a quasi-isomorphism on such crossings:

  • Φ : C
  • ⊗ F2

∼

− → C

  • ⊗ F2

. In particular, Φ is non-trivial.

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Review on Khovanov homology

1

Introduction

2

Review on Khovanov homology As a bigraded module On differential

3

Vassiliev skein relation on Khovanov homology

4

Main Results

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As a bigraded module

D: a diagram of an (ordinary) link. Definition A state on D is a smoothng of D on all its crossings:

0-smoothing

← − − − − − − − c

1-smoothing

− − − − − − − → . a state Ds is topologically of the form Ds ∼ = (S1)∐π0(Ds). Notation For a state Ds, write |s| the number of 1-smoothings.

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As a bigraded module

Definition For a link diagram D, we set Ci,j(D) :=

  • Ds:state

|s|=i

(V ⊗π0(Ds))j−i , where V := Z{1, x} is a graded abelian group with deg 1 = 1 and deg x = −1. Basis Ci,j(D) is a free abelian group generated by enhanced states. Example An enhanced state on trefoil 31: x 1 ∈ C1,1    

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On differential

Observation V underlies a (1+1)-TQFT (in the sense of [Atiyah, 1988]) associated to the Frobenius algebra Z[x]/(x2). The saddle operation assigns 1-smoothings to 0-smoothings: : → δc : Ci,j(D) → Ci+1,j(D) for each crossing c: p → p(1) p(2) p q → pq

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On differential

Definition

1 For a link diagram D, put

d :=

  • ±δc : Ci,j(D) → Ci+1,j(D)

, where the sum is taken so that δc apply on all the 0-smoothings.

2 n+, n−: the numbers of positive and negative crossings.

Ci,j(D) := Ci+n−,j+2n−−n+(D) . C(D) = (C∗,⋆(D), d) is called the Khovanov complex of D. Theorem ([Khovanov, 2000]) For every abelian group M, the homology group Khi,j(L; M) := Hi(C∗,j(D) ⊗ M) is independent of the choice of a diagram D of L.

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Vassiliev skein relation on Khovanov homology

1

Introduction

2

Review on Khovanov homology

3

Vassiliev skein relation on Khovanov homology Mapping cone as subtraction Genus-1 map Invariance under elementary moves Cube of resolutions Example

4

Main Results

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Mapping cone as subtraction

Question How can we realize the “subtraction” Kh

  • “−” Kh
  • ?

A Consider a map f : C

  • → C
  • and take the mapping cone Cone(f).

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Mapping cone as subtraction

Definition Let f : X → Y be a chain map. Define a complex Cone(f) by Cone(f)i := Y i ⊕ Xi+1 with differential d = dY f −dX

  • : Y i ⊕ Xi+1 → Y i+1 ⊕ Xi+2

. Lemma If f : X → Y is a chain map, then the sequence below is exact · · · → Hn(X)

f∗

− → Hn(Y )

i∗

− → Hn(Cone(f))

p∗

− → Hn+1(X) → . . . i : Y → Cone(f) is the canonical inclusion; p : Cone(f) → X∗+1 is the canonical projection.

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Genus-1 map

Proposition The 2-dimensional cobordism : → gives rise to a chain map

  • Φ : C
  • ⊗ F2 → C
  • ⊗ F2

. Remark As a realization of crossing change, Φ is not the standard one. Indeed, the standard one is of degree = (0, 0).

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Genus-1 map

Presentation

  • Φ

: C

  • ⊗ F2

→ C

  • ⊗ F2

p → p q → (pq)(1) (pq)(2) p q → p → p(1)p(2)

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Invariance under elementary moves

Proposition The chain map Φ is invariant under elementary moves of double points. Elementary moves [Bataineh-Elhamadi-Hajij-Youmans, 2018] ΩIVa ← → ΩIVe ← → ΩV ← →

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Invariance under elementary moves

Sketch of the proof of Proposition Show that the genus-1 map Φ : C

  • → C
  • is invariant instead of

Φ is. Invariance is verified as the commutativity of appropriate diagrms: e.g. Ci,j     Ci,j

  • Ci,j

    Ci,j−2     Ci,j−2

  • Ci,j−2

   

Φ RII RII Φ RII RII

.

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Cube of resolutions

Definition Let D be a diagram of a singular link.

1 c#(D): the set of double points in D. 2 A resolution of D is a diagram Dr without double points obtained

by resolving double points of D: b−

−-resolution

← − − − − − − − b

+-resolution

− − − − − − − → b+ .

3 |r| the number of −-resolutions. 24 / 31

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Cube of resolutions

Observation Resolutions of D

1:1

← → subsets of c#(D). If Dr− and Dr+ differ at only one double point, then

  • Φ : C(Dr−) ⊗ F2 → C(Dr+) ⊗ F2

. The maps Φ at different double points commute with each other. Lemma For each singular link diagram D, we have a c#(D)-cube of Khovanov complexes of resolutions of D and the genus-1 maps. Definition We denote by C∗,⋆(D; F2) the multiple mapping cone of the c#(D)-cube.

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Example

Cube for a b : C   a− b−   ⊗ F2

  • Φa

− − → C   a+ b−   ⊗ F2

  • Φb ↓

↓

Φb

C   a− b+   ⊗ F2

  • Φa

− − → C   a+ b+   ⊗ F2

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Main Results

1

Introduction

2

Review on Khovanov homology

3

Vassiliev skein relation on Khovanov homology

4

Main Results Khovanov homology on singular links Example: “figure-eight” FI-relation

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Khovanov homology on singular links

Theorem (Ito-Y. arXiv:1911.09308) Let L be a singular link and D a diagram.

1 The homology Khi,j(L; F2) := Hi(C∗,j(D; F2)) is independent of

the choice of D.

2 If L has no double point, then Khi,j(L; F2) agrees with the

  • rdinary Khovanov homology (with coefficients in F2).

3 For each double point of L, there is a long exact sequence

· · · → Khi,j

  • ; F2
  • Φ∗

− − → Khi,j

  • ; F2
  • → Khi,j
  • ; F2
  • → Khi+1,j
  • ; F2
  • → . . .

.

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Example: “figure-eight”

Khi,j        

  • Φ

− → Khi,j         → Khi,j         Proposition The below is the table of the homology groups on the middle (resp. on the right): i\j −5 −4 −3 −2 −1 1 2 3 4 5 −2 F2 F2 −1 F2 F2 F

×

F

×

1 F2 F2 2 F2 F2

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FI-relation

Theorem (Ito-Y. arXiv:1911.09308) Khovanov homology (with coefficients in F2) satisfies FI-relation; i.e. Kh

  • ; F2
  • = 0

. Proof Direct computation shows the diagram below commutes: C

  • ; F2
  • C
  • ; F2
  • C
  • ; F2
  • RI−

RI+

  • Φb

.

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Reference I

[Atiyah, 1988] Atiyah, M. (1988). Topological quantum field theories. Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 68(1):175–186. [Bataineh et al., 2018] Bataineh, K., Elhamdadi, M., Hajij, M., and Youmans, W. (2018). Generating sets of reidemeister moves of oriented singular links and quandles. Journal of Knot Theory and Its Ramifications, 27(14):1850064–1–15. [Birman and Lin, 1993] Birman, J. S. and Lin, X.-S. (1993). Knot polynomials and Vassiliev’s invariants. Inventiones Mathematicae, 111(2):225–270. [Khovanov, 2000] Khovanov, M. (2000). A categorification of the Jones polynomial. Duke Mathematical Journal, 101(3):359–426.