When the Theories Meet: Khovanov Homology as Hochschild Homology of Links

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 comultiplicationfree version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong and extended here to noncommutative algebras. 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. 1. Hochschild homology and cyclic homology We recall in this section definition of Hochschild homology and cyclic homology and we sketch two classical calculations for tensor algebras and symmetric tensor algebras. More calculations are reviewed in Section 4 in which we use our main result, Theorem 3.1, to obtain new results in Khovanov homology, in particular solving some conjectures from [H-P-R]. We follow [Lo] in our exposition of Hochschild homology. Let k be a commutative ring and A a k-algebra (not necessarily commutative). Let M be a bimodule over A that is a k-module on which A operates linearly on the left and on the right in such a way that (am)a′ = a(ma′) for a, a′ ∈ A and m ∈ M. The actions of A and k are always compatible (e.g. m(λa) = (mλ)a = λ(ma)). When A has a unit element 1 we always assume that 1m = m1 = m for all m ∈ M. Under this unital hypothesis, the bimodule M is equivalent to a right A⊗Aop-module via m(a′ ⊗ a) = ama′. Here Aop denotes the opposite algebra of A that is A and Aop are the same as sets but the product a · b in Aop is the product ba in A. The product map