University of Oxford. http://www.maths.ox.ac.uk/
Combinatorial theory seminar. November 9, 2008.

Title: Graphs on surfaces and virtual knots.
	
Abstract: 
Regions of a link diagram can be colored in black and white in a
checkerboard manner. Putting a vertex in each black region and connecting
two vertices by an edge if the corresponding regions share a crossing
yields a planar graph. In 1987 Thistlethwaite proved that the Jones
polynomial of the link can be obtained by a specialization of the Tutte
polynomial of this planar graph. The goal of my talk will be an explanation of 
a generalization of Thistlethwaite's theorem to virtual links. In this case 
graphs will be embedded into a (higher genus, possibly non-oriented) surface. 
For such graphs we used a generalization of the Tutte polynomial called the 
Bollobas-Riordan polynomial. For graphs on surfaces the natural duality can be
generalized to a duality with respect to a subset of edges. The generalized dual 
graph might be embedded into a different surface. I will explain a relation 
between the Bollobas-Riordan polynomials of dual graphs. This relation unifies 
various Thistlethwaite's type theorems.