Seminars "Combinatorics of Vassiliev invariants". Speaker: Grisha Taroyan

One can functorially associate a bigraded complex C(D) of abelian groups to a planar diagram D of a link L. Complex C(D) can be shown to be independent up quasi-isomorphism from a specific planar projection of the link L. Thus homology groups H(L) of the complex C(L) yield a new invariant of the link L known as Khovanov homology. A celebrated result of Khovanov is the following 

Theorem. The graded Poincare polynomial Kh(L)(t,q)=Σ t^i q^j dim H^{i,j}(L) is a link invariant which specifies to the unnormalized Jones polynomial of the link L at t=-1.
The bulk of my talk will be dedicated to proving invariance of C(L) under Reidemeister moves of planar diagrams. Then I will prove the theorem stated above. Time permitting I will also talk about functoriality of H(L) with respect to link cobordism and Khovanov homology as 4-dimensional TQFT. 

The talk follows several papers. The first part can be deduced from [1]. The original paper of Khovanov [2] is also an invaluable source in understanding the theory although a slightly more difficult and unpolished one. Finally, Jacobsson's paper provides a necessary background and proof of Khovanov homology's invariance under link cobordism. Definition of 4-dimensional TQFT can be found in standard sources like the following entry on nLab [4]. 

References: [1] Bar-Natan, Dror. “On Khovanov’s categorification of the Jones polynomial.” Algebraic & Geometric Topology 2.1 (2002): 337-370. 

[2] Khovanov, Mikhail. “A categorification of the Jones polynomial.” Duke Mathematical Journal 101.3 (2000): 359-426. 

[3] Jacobsson, Magnus. “An invariant of link cobordisms from Khovanov homology.” Algebraic \& Geometric Topology 4.2 (2004): 1211-1251. 

[4] nLab authors. “4d-TQFT.” (November 2021).

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