Combinatorial patterns in Khovanov type graph homology motivated by Hochschild homology

Jozef H. Przytycki

Knots in Washington XXII
May 5-7, 2006
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), przytyck@gwu.edu, Yongwu Rong (GWU), rong@gwu.edu, Alexander Shumakovitch (GWU), shurik@gwu.edu

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Combinatorial patterns in Khovanov type graph homology motivated by Hochschild homology
by
Jozef H. Przytycki
George Washington University
Coauthors: Milena Pabiniak (GWU) and Radmila Sazdanovic (GWU)

The algebra of truncated polynomials A_m = Z[x]/(x^m) plays an important role in the theory of Khovanov and Khovanov-Rozansky homology of links. It is not difficult to compute Hochschild homology of A_m and the only torsion, Z_m, appears in grading (i, [(m(i+1))/2]) for any odd i. We have demonstrated that Hochschild homology is closely related to Khovanov homology via comultiplication free graph homology. We analyze here grading of graph cohomology which is producing torsion for a polygon. We find completely the cohomology H^{1, v-1}_A₂(G) and H^{1, 2v-3}_A₃(G). In the following talks the case H^{1, 2v-3}_A₃(G) is described by my coauthors. Here we compute the first cohomology of a graph with underlying algebra A₂. We notice that working with the dual chain complex is much easier to visualize. In particular, we show that H₀^A₂(G) is a "symmetric" homology of a graph, that is the boundary of an edge is equal the sum of its endpoints.

Date received: May 5, 2006