doi:10.1016/j.ejc.2006.12.004 
Copyright © 2007 Elsevier Ltd All rights reserved.
Knot invariants and the Bollobás–Riordan polynomial of embedded graphs
Iain Moffatt1, a, 
aDepartment of Applied Mathematics, Charles University, Malostranské nam. 25, 118 00 Praha 1, Czech Republic
Received 30 May 2006;
accepted 12 December 2006.
Available online 20 January 2007.
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Abstract
For a graph G embedded in an orientable surface Σ, we consider associated links 
in the thickened surface Σ×I. We relate the HOMFLY polynomial of 
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.
Fig. 1. Constructing the link 
.
Fig. 2. An embedded graph and associated link projections.
Fig. 3. Local differences in links.
Fig. 4. Tangles associated to edge weights.
Fig. 5. Constructing the medial link.
Abstract
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