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

Abstract

We show that Khovanov homology and Hochschild homology theories share a common structure. In fact they overlap: Khovanov homology of the (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 the $\mathbb{M}$-reduced case, and in the case of a polygon extended to noncommutative algebras. In this framework we prove that for any unital algebra $\mathcal{A}$ the Hochschild homology of $\mathcal{A}$ is isomorphic to graph cohomology over $\mathcal{A}$ 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.