Iain Moffatt

The chromatic polynomial of fatgraphs and its categorification

Abstract 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 this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobas–Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.

Knot invariants and the Bollobás-Riordan polynomial of embedded graphs

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

Partial duality and Bollobás and Riordan's ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobas and Riordans 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 and I give a simple proof of the relation which used the homfly polynomial of a knot.

Unsigned State Models for the Jones Polynomial

It is a well-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 process of deriving this result, we show that for every diagram of a link in S3 there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating to the Jones and Bollobás-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.

Graphs on Surfaces

1. Embedded Graphs .- 2. Generalised Dualities .- 3. Twisted duality, cycle family graphs, and embedded graph equivalence .- 4. Interactions with Graph Polynomials .- 5. Applications to Knot Theory .- References .- Index .

Partial duals of plane graphs, separability and the graphs of knots

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.

A characterization of partially dual graphs

In this article, 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 correspondence, and gives a combinatorial characterization of partial duality.

Separability and the genus of a partial dual

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 embedded graph give rise to a partial dual of a given genus. This problem turns out to be intimately connected to the separability of the embedded graph. We determine how separability is related to the genus of a partial dual. We use this to characterize partial duals of graphs embedded in the plane, and in the real projective plane, in terms of a particular type of separation of an embedded graph. These characterizations are then used to determine a local move relating all partially dual graphs in the plane and in the real projective plane.

A Penrose polynomial for embedded graphs

We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which cannot be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable chequerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G. This allows us to obtain a new reformulation of the Four Colour Theorem.

Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.