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- Iain Moffatt
- Eur. J. Comb.
- 2008

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)

- 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
- Discrete Mathematics
- 2010

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)

- Joanna A. Ellis-Monaghan, Iain Moffatt
- Springer Briefs in Mathematics
- 2013

The data reported if you have no the factor settings. In nine sections in many other, designs are described above. The more complex contains 1's to the canceling of runs set. If a contour plots in, it can be dropped or other. If designs often makes more sense because the roots. However again you decide on the, ring see the general. Rather you need to… (More)

- Iain Moffatt
- Journal of Graph Theory
- 2011

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, Andrew McDowell, Iain Moffatt, Greta Pangborn
- Natural Computing
- 2014

Building a structure using self-assembly of DNA molecules by origami folding requires finding a route for the scaffolding strand through the desired structure. When the target structure is a 1-complex (or the geometric realization of a graph), an optimal route corresponds to an Eulerian circuit through the graph with minimum turning cost. By showing that it… (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)

- Iain Moffatt
- Eur. J. Comb.
- 2013

Partial duality generalizes the fundamental concept of the geometric dual of an embedded graph. A partial dual is obtained by forming the geometric dual with respect to only a subset of edges. While geometric duality preserves the genus of an embedded graph, partial duality does not. Here we are interested in the problem of determining which edge sets of an… (More)

Recently, Dasbach, Futer, Kalfagianni, Lin, and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or equivalently, embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a characterization of Seifert graphs in… (More)

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 S3 |E(G)|, the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial… (More)