A 2-category of chronological cobordisms and odd Khovanov homology

Abstract

Abstract. We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms along any 3-dimensional subcube is trivial. This allows us to create a chain complex, whose homotopy type modulo certain relations is a link invariant. Both the original and the odd Khovanov homology can be recovered from this construction by applying certain strict 2-functors. We describe other possible choices of functors, including the one that covers both homology theories and another generalizing dotted cobordisms to the odd setting. Our construction works as well for tangles and is conjectured to be functorial up to sign with respect to tangle cobordisms.

On a triply graded Khovanov homology

“…The category 2Cob of 2-dimensional cobordisms is usually considered as Z-graded, with the degree function given by the Euler characteristic of a cobordism. It was shown in [Put08,Put13] that this degree can be split into two numbers, one counting merges and births, whereas the other splits and deaths: Indeed, the only relations that affect the set of critical points either create or remove a pair birth-merge or split-death, which does not change the two numbers. Because of that one can try to enhance the construction of Khovanov homology H i,j (L) [Kho99] to a triply graded homology H(L) i,p,q with H i,j (L) = p+q=j H i,p,q (L).…”

“…Unfortunately, one does not obtain a stronger invariant in this way, as after a normalization H(L) i,p,q = 0 unless p = q. This is the reason why the author dropped this idea in his earlier works [Put08,Put13] on unification of the Khovanov homology with its odd variant [ORS13].…”

“…We begin with a brief description of the category of chronological cobordisms kChCob and graded tensor categories, introduced in [Put13], which provide a framework for the construction of the generalized Khovanov homology H(L). The construction of the generalized Khovanov complex Kh(D) is presented in Section 3 following [Put13]. The only difference is in regarding it as an object graded by Z × Z, instead of graded by integers as in the original construction.…”

“…The only difference is in regarding it as an object graded by Z × Z, instead of graded by integers as in the original construction. The section ends with a proof that all the homotopy equivalences used in [Put13] to prove invariance of Kh(D) under Reidemeister moves preserve the new grading. Then in Section 4 we show that the new grading does not lead to new invariants.…”

Knot invariants arising from homological operations on Khovanov homology

“…They are constructed using a link homology H ξ (L) that is defined over a ring Z ξ := Z[ξ]/(ξ 2 −1) and unifies the even and odd Khovanov homology theories. H e (L) and H o (L) can be recovered from H ξ (L) by taking coefficients in certain modules over Z ξ [Pu13]. Namely, there are isomorphisms of graded abelian groups…”