On Link Homology Theories from Extended Cobordisms

TL;DR

This paper develops extended quantum field theories based on embedded cobordisms and 2-categories, unifying various Khovanov-type link homology theories and establishing conditions for their equivalence.

Contribution

It introduces a new framework of extended quantum field theories that generalizes existing link homology theories, including Khovanov homology, via embedded cobordisms and 2-categories.

Findings

Khovanov and odd Khovanov homologies fit into the extended framework.

Any EQFT with a Z_2-extension matching Khovanov mod 2 yields a homology equivalent to Khovanov.

The framework unifies multiple link homology theories under a common categorical setting.

Abstract

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel-Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any EQFT based on a Z_2-extension of the embedded cobordism category which coincides with Khovanov after reducing the coefficients modulo 2, gives rise to a link invariant homology theory isomorphic to those of Khovanov.