Department of Mathematics
From Khovanov homology to Hochschild homology and back in 50 Minutes
Jozef Przytycki
George Washington University, Washington D. C.
We start this talk by describing the Tait construction of link diagrams from signed plane graphs, and conversely, the construction of signed plane graphs from link diagrams. In 1999 M. Khovanov introduced a homology theory which categorifies the Jones polynomial of links. We use the Tait construction to argue that one can understand Khovanov homology of links by describing first graph homology. Hochschild homology is the older theory, developed in 1945 to analyze rings and algebras. We show that Khovanov homology and Hochschild homology share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as 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 Y. Rong and L. Helme-Guizon. 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 observation (that two theories meet) will encourage a flow of ideas in both directions betwewen Hochschild/cyclic and Khovanov homology theories.
All interested persons are welcome.