The chromatic polynomial of fatgraphs and its categorification

Abstract

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobas-Riordan polynomial (a generalisation of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.

A substantial revision. To appear in Advances in Mathematics

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Cited by in corpus (12)

• Knot invariants and the Bollobas-Riordan polynomial of embedded graphs
• Partial duality and Bollobas and Riordan's ribbon graph polynomial
• Unsigned state models for the Jones polynomial
• Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
• Topological Tutte Polynomial
• Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial
• On mutation and Khovanov homology
• The 2-Factor Polynomial Detects Even Perfect Matchings
• On the combinatorics of sparsification
• Uniform generation of RNA pseudoknot structures with genus filtration
• A bijection between unicellular and bicellular maps
• On the HOMFLY and Tutte polynomials