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For a graph G embedded in an orientable surface Σ, we consider associated links L(G) in the thickened surface Σ × I. We relate the HOMFLY polynomial of L(G) 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… (More)

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan's ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory… (More)

- David M Jackson, Iain Moffatt, Alejandro Morales
- 2005

We study the effect of Feynman integration and diagrammatic differential operators on the structure of group-like elements in the algebra generated by coloured vertex-oriented uni-trivalent graphs. We provide applications of our results to the study of the LMO invariant, a quantum invariant of manifolds. We also indicate further situations in which our… (More)

- Martin Loebl, Iain Moffatt
- 2008

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply… (More)

- Iain Moffatt
- 2009

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the… (More)

In this paper, we extend the recently introduced concept of partially dual ribbon graphs to graphs. We then go on to characterize partial duality of graphs in terms of bijections between edge sets of corresponding graphs. This result generalizes a well known result of J. Edmonds in which natural duality of graphs is characterized in terms of edge… (More)

- Joanna A Ellis-Monaghan, Iain Moffatt
- 2011

We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge, and taking the partial dual with respect to the edge. These two operations give rise to an action of S 3 |E(G)| , the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial… (More)

- Stephen Huggett, Iain Moffatt
- 2007

We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobás-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we… (More)

We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, L(G), and of and Bollobás and Riordan, R(G), as well as for the classical Tutte polynomial T (G). For example, we express R(G) and T (G) as a sum of chromatic polynomials, show that R(G) counts non-crossing graph… (More)