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. I will explain a generalization of this 
theorem to virtual links. In this case the graph will be a ribbon graph, 
which means that it will be embedded into a (higher genus, possibly 
non-oriented) surface. For such graphs we use a generalization of the 
Tutte polynomial discovered recently by B. Bollobas and O. Riordan. This 
is a joint work with Jeremy Voltz.