Knot invariants and the Bollobás–Riordan polynomial of embedded graphs

Abstract

For a graph $G$ embedded in an orientable surface $\Sigma$, we consider associated links $\mathcal{L}\left(G\right)$ in the thickened surface $\Sigma \times I$. We relate the HOMFLY polynomial of $\mathcal{L}\left(G\right)$ to the recently defined Bollobás–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 Bollobás–Riordan polynomial. We then consider the specialization to the Jones polynomial and recent results of Chmutov and Pak to relate the Bollobás–Riordan polynomials of an embedded graph and its tensor product with a cycle.