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

We show that Khovanov homology and Hochschild homology theories share commonstructure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link canbe interpreted as a Hochschild homology of the algebra underlining the Khovanovhomology. In the classical case of Khovanov homology we prove the concreteconnection. In the general case of Khovanov-Rozansky, $sl(n)$, homology andtheir deformations we conjecture the connection. The best framework to exploreour ideas is to use a comultiplication-free version of Khovanov homology forgraphs 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 agraph 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 bothdirections between Hochschild/cyclic homology and Khovanov homology theories.