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 relation on Khovanov homology Main Results Reference Contents Vassiliev skein relation on 3 Introduction 1 Khovanov homology Goal of the talk Mapping cone as subtraction Background: Vassiliev Genus- 1 map invariants Invariance under elementary Backgound: categorification moves Motivating Question Cube of resolutions Goal of the talk (continued) Example Main Theorem 1 Main Results 4 Main Theorem 2 Khovanov homology on Review on Khovanov homology singular links 2 As a bigraded module Example: “figure-eight” On differential FI-relation 2 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Introduction Introduction 1 Goal of the talk Background: Vassiliev invariants Backgound: categorification Motivating Question Goal of the talk (continued) Main Theorem 1 Main Theorem 2 Review on Khovanov homology 2 Vassiliev skein relation on Khovanov homology 3 Main Results 4 3 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference 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... 4 / 31
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 S 3 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] Taylor expansion Quantum Vassiliev Jones ∈ Invariants Invariants polynomial 5 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Backgound: categorification Khovanov homology Kh i,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. � ( − 1) i q j dim Kh i,j ( L ) ∈ Z [ q, q − 1 ] [ Kh ( L )] q := , i,j 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 ) . 6 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Motivating Question ??? ??? Kh decategorify categorify ??? Taylor expansion Vassiliev V Invariants Question 1 How does Khovanov homology relate to Vassiliev invariants? 2 Furthermore, does it produces categorifications of Vassiliev invariants ? 7 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference 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 · · · → H n ( X ) → H n ( Y ) → H n ( Z ) → H n +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 . 8 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Main Theorem 1 Theorem (Ito-Y. arXiv:1911.09308) Khovanov homology (with coefficients in F 2 ) extends to a singular link invariant so that there is a long exact sequence � � � � � Φ ∗ · · · → Kh i,j → Kh i,j ; F 2 − − ; F 2 � � � � → Kh i,j → Kh i +1 ,j ; F 2 ; F 2 → . . . . 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. 9 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Main Theorem 1 Corollary Khovanov homology categorifes unnormalized Jones polynomial even on singular links. Proof Observe that the identity � � �� � � �� � � �� ; F 2 − ; F 2 + ; F 2 = 0 Kh Kh Kh q q q arising from the long exact sequence is exactly the Vassiliev skein relation for unnormalized Jones polynomial. 10 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Main Theorem 2 Theorem (Ito-Y. arXiv:1911.09308) Khovanov homology (with coefficients in F 2 ) satisfies FI-relation ; i.e. � � Kh ; F 2 = 0 . Equivalently The genus- 1 map � Φ is a quasi-isomorphism on such crossings: � � � � ∼ � Φ : C ⊗ F 2 − → C ⊗ F 2 . � In particular, � Φ is non-trivial . 11 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Review on Khovanov homology Introduction 1 Review on Khovanov homology 2 As a bigraded module On differential Vassiliev skein relation on Khovanov homology 3 Main Results 4 12 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference 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: c 0 -smoothing 1 -smoothing ← − − − − − − − − − − − − − − → . � a state D s is topologically of the form D s ∼ = ( S 1 ) ∐ π 0 ( D s ) . Notation For a state D s , write | s | the number of 1 -smoothings . 13 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference As a bigraded module Definition For a link diagram D , we set � ( V ⊗ π 0 ( D s ) ) j − i C i,j ( D ) := , D s :state | s | = i where V := Z { 1 , x } is a graded abelian group with deg 1 = 1 and deg x = − 1 . Basis C i,j ( D ) is a free abelian group generated by enhanced states . Example ∈ C 1 , 1 An enhanced state on trefoil 3 1 : x 1 14 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference On differential Observation V underlies a (1+1)-TQFT (in the sense of [Atiyah, 1988]) associated to the Frobenius algebra Z [ x ] / ( x 2 ) . The saddle operation assigns 1 -smoothings to 0 -smoothings: : → � δ c : C i,j ( D ) → C i +1 ,j ( D ) for each crossing c : p p (1) p (2) �→ p pq �→ q 15 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference On differential Definition 1 For a link diagram D , put � ± δ c : C i,j ( D ) → C i +1 ,j ( D ) d := , where the sum is taken so that δ c apply on all the 0 -smoothings. 2 n + , n − : the numbers of positive and negative crossings. C i,j ( D ) := C i + n − ,j +2 n − − 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 Kh i,j ( L ; M ) := H i ( C ∗ ,j ( D ) ⊗ M ) is independent of the choice of a diagram D of L . 16 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Vassiliev skein relation on Khovanov homology Introduction 1 Review on Khovanov homology 2 Vassiliev skein relation on Khovanov homology 3 Mapping cone as subtraction Genus- 1 map Invariance under elementary moves Cube of resolutions Example Main Results 4 17 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference 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 ) . 18 / 31
Introduction Review on Khovanov homology Vassiliev skein relation on Khovanov homology Main Results Reference Mapping cone as subtraction Definition Let f : X → Y be a chain map. Define a complex Cone( f ) by Cone( f ) i := Y i ⊕ X i +1 with differential � d Y � f : Y i ⊕ X i +1 → Y i +1 ⊕ X i +2 d = . 0 − d X Lemma If f : X → Y is a chain map, then the sequence below is exact f ∗ i ∗ p ∗ · · · → H n ( X ) → H n ( Y ) → H n (Cone( f )) → H n +1 ( X ) → . . . − − − i : Y → Cone( f ) is the canonical inclusion; p : Cone( f ) → X ∗ +1 is the canonical projection. 19 / 31
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