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Organizers |
A Khovanov-type homology theory for graphs
by
Laure Helme-Guizon
George Washington University
Coauthors: Yongwu Rong, George Washington University, Józef Przytycki, George Washington University, Mikhail Khovanov, UC Davis.
In recent years, there has been a great deal of interests in Khovanov homology theory. For each link L, Khovanov defines a family of homology groups whose "graded" Euler characteristic is the Jones polynomial of L. These groups were constructed through a categorification process which starts with a state sum of the Jones polynomial, constructs a group for each term in the summation, and then defines boundary maps between these groups appropriately for each positive integer n.
It is natural to ask if similar categorifications can be done for other invariants with state sums.
In the first part of this talk, we establish a homology theory that categorifies the chromatic polynomial for graphs. We show our homology theory satisfies a long exact sequence which can be considered as a categorification for the well-known deletion-contraction rule for the chromatic polynomial. This exact sequence helps us to compute the homology groups of several classes of graphs. In particular, we point out that torsions do occur in the homology for some graphs. This is joint work with Yongwu Rong, George Washington University.
The second part of this talk will discuss for which graphs these homology groups have torsion. This is joint work with Yongwu Rong and Józef Przytycki, George Washington University.
The third part of this talk will discuss how to extend this construction to other algebras. This is joint work with Yongwu Rong, George Washington University and Mikhail Khovanov, UC Davis.
Date received: February 6, 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 # capo-24.