Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

Details Knot invariants and the Bollobas-Riordan polynomial of embedded graphs Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

2006-05-17

arxiv.org/abs/math/0605466v2

European J. Combin., 29 (2008) 95-107

Mathematics

Combinatorics

v2: minor corrections, to appear in European Journal of Combinatorics

Scientific paper

10.1016/j.ejc.2006.12.004

For a graph G embedded in an orientable surface \Sigma, we consider associated links L(G) in the thickened surface \Sigma \times I. We relate the HOMFLY polynomial of L(G) to the recently defined Bollobas-Riordan polynomial of a ribbon graph. This generalizes celebrated results of Jaeger and Traldi. We use knot theory to prove results about graph polynomials and, after discussing questions of equivalence of the polynomials, we go on to use our formulae to prove a duality relation for the Bollobas-Riordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the Bollobas-Riordan polynomials of an embedded graph and its tensor product with a cycle.

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