Generalized duality for graphs on surfaces and the signed Bollobás--Riordan polynomial

Authors:
Sergei Chmutov
Affiliations:
Department of Mathematics, The Ohio State University, Mansfield, 1680 University Drive, Mansfield, OH 44906, United States
Venue:
Journal of Combinatorial Theory Series B
Year:
2009

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Abstract

We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobas-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobas-Riordan polynomials.