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Confluence of Khovanov homology and Hochschild homology: application to truncated polynomial algebra
by
Jozef H. Przytycki
George Washington University

Half a year ago (precisely May 4, 2005) I noted 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 comultiplication-free version of Khovanov homology for graphs developed by L. Helme-Guizon and Y. Rong. In this framework we prove that for any commutative unital algebra A and A-bimodule M the Hochschild homology of A with coefficients in M is isomorphic to M-reduced graph homology over A of a polygon. In this talk we show how one can extend the results from a polygon to some other graphs. This part is mostly speculative, and we concentrate on truncated polynomial algebra A_m = Z[x]/(x^m). We conjecture, in particular, that if a simple graph G contains a triangle than H_Am^{1, (m−1)(v−1) − (m−2)}(G) contains Z_m. The last part of the talk describes joint work with U.Kaiser, M. Pabiniak, R. Sazdanovic and A. Shumakovitch.

Paper reference: arXiv:math.GT/0402402, arXiv:math.GT/0507245, arXiv:math.GT/0509334

Date received: December 5, 2005

Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # carw-16.