On link homology theories from extended cobordisms

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 refine2-dimensional cobordisms by taking into account their embedding into R 3 . 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 (strict) EQFT based on a Z2-extension of the embedded cobordism category that coincides with Khovanov homology after reducing the coefficients modulo 2 gives rise to a link invariant homology theory isomorphic to that of Khovanov.