Categorification of the Dichromatic Polynomial for Graphs

Details Categorification of the Dichromatic Polynomial for Graphs Categorification of the Dichromatic Polynomial for Graphs

2005-04-12

arxiv.org/abs/math/0504239v2

Mathematics

Geometric Topology

12 pages, 2 figures; Section 3 corrected, added Section 5

Scientific paper

For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations of the Tutte polynomial of graphs. Also, for each graph and integer $n\le 2$, we define the different one variable $n$-specializations of the dichromatic polynomials, and for each polynomial we define graded chain complex whose graded Euler characteristic is equal to that polynomial. Furthermore, we explicitly categorify the specialization of the Tutte polynomial for graphs which corresponds to the Jones polynomial of the appropriate alternating link.

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