New Categorifications of the Chromatic and the Dichromatic Polynomials for Graphs
Abstract
In this paper, for each graph G, we define a chain complex of graded modules over the ring of polynomials, whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules, whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of HOMFLYPT polynomial. We also give simplified definition of this triply-graded link homology theory.
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