Title: Graphs on surfaces via planar graphs


Abstract.
Graphs on surfaces can be studied in terms of plane graphs via their projections 
preserving the rotation system. For non-planar graphs such a projection will 
have singularities. The simplest singularities are double points on edges of 
the graph. Using them we supplement the image of the graph with some additional 
edges and vertices. Thus we obtain a relative plane graph which is a plane graph 
with a distinguished subset of edges.
    The Tutte polynomial is a famous invariant of the graph. It was generalized 
in topological setting for graphs on surfaces by B.Bollobas and O.Riordan and 
for relative plane graphs by Y.Diao and G.Hetyei. We found a relation between 
these polynomials for graphs obtained by the construction above. 
     This relation has an application in knot theory. The classical Thistlethwaite 
theorem to relates the Jones polynomial of a link to the Tutte polynomial of a 
plane graph obtained from a checkerboard coloring of the regions of the link 
diagram. Our relation conforms two generalizations of the Thistlethwaite theorem 
to virtual links.
     This is a joint work with Clark Butler.