Topology Atlas | Conferences

A categorification for the Tutte polynomial for matroids: From a matroid M, we produce a chain complex whose graded Euler Characteristic is the Tutte polynomial of M.
by
Laure Helme-Guizon
George Washington University

In recent years, there has been a great deal of interest 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.

It is natural to ask if similar categorifications can be done for other invariants with state sums.

In 2004, Yongwu Rong and I established a homology theory which categorifies the chromatic polynomial for graphs.

Then, Yongwu Rong and Fanny Jasso-Hernandez came up with a categorification for the Tutte polynomial for graphs. This works really differs from ours since the graded Euler characteristic is a two-variable polynomial.

Matroids are a generalization of graphs (and of vectors spaces) so it is natural to ask if a categorification can be obtained for the Tutte polynomial for matroids.

In this talk, I will remind you briefly of what a matroid is (if there is any popular demand/need for that) and then I will show you a categorification for the Tutte polynomial for matroids. It is more than a cosmetic upgrade of the categorification for the Tutte polynomial for graphs by Yongwu Rong and Fanny Jasso-Hernandez since their construction relies heavily on the notion of connected components of a graph which doesn’t exist for matroids.

I wish to thank Fanny Jasso-Hernandez and Joe Bonin (both from The George Washington University) for helpful discussions.

Date received: November 12, 2006

Copyright © 2006 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 # catp-10.