Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

Abstract

For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ\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.

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

References in corpus (1)

• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial

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• Non-orientable quasi-trees for the Bollobas-Riordan polynomial
• Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
• Graph polynomials and their applications II: Interrelations and interpretations
• Arrow ribbon graphs
• Types of embedded graphs and their Tutte polynomials
• A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan
• Hopf algebras and Tutte polynomials
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• Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial