When the theories meet: Khovanov homology as Hochschild homology of links

Abstract

We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connection. In the general case of Khovanov-Rozansky, $sl(n)$, homology and their deformations we conjecture the connection. The best framework to explore our ideas is to use a comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to to $\mathbb M$-reduced case, and to noncommutative algebras (in the case of a graph being a polygon). In this framework we prove that for any unital algebra $\A$ the Hochschild homology of $\A$ is isomorphic to graph homology over $\A$ of a polygon. We expect that this paper will encourage a flow of ideas in both directions between Hochschild/cyclic homology and Khovanov homology theories.