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 M-reduced case, and in the case of a polygon extended to noncommutative algebras. In this framework we prove that for any unital algebra A the Hochschild homology of A is isomorphic to graph cohomology 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.