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Organizers |
A Khovanov-type cohomology theory for graphs
by
Laure Helme-Guizon
GWU
Coauthors: Y. Rong and J. Przytycki, GWU
In recent years, there have 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, Washington DC, USA
The second part of this talk will discuss for which graphs these homology groups have torsion and provide some computational results. This is joint work with Yongwu Rong and Józef Przytycki, George Washington University, Washington DC, USA
Paper references:
[1] A categorification for the chromatic polynomial, LHG, Yongwu Rong : arXiv:math.CO/0412264 or http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-53.abs.html
[2] Graph Cohomologies from Arbitrary Algebras, LHG, Yongwu Rong arXiv:math.QA/0506023
[3] LHG, Y. Rong and J. Przytycki, Torsion in Graph Homology, arXiv:math.GT/0507245
Paper reference: arXiv:math.CO/0412264, arXiv:math.QA/0506023, arXiv:math.GT/0507245
Date received: December 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 # carw-18.