November 30, 2010

Polynomials of graphs on surfaces

Abstract.
The Jones polynomial of links in 3-space is a specialization of the Tutte 
polynomial of corresponding plane graphs. There are several generalizations 
of the Tutte polynomial to graphs embedded into a surface. Some of them are 
related to the theory of virtual links. Although virtual link theory predicts 
some relations between these generalizations. I will report about the results 
obtained in this direction during my summer program "Knots and Graphs".

In particular I will compare three polynomials of graphs on surfaces and a 
relative version of the Tutte polynomial of planar graphs. The first polynomial, 
defined by M.Las Vergnas, uses a strong map of the bond matroid of the dual 
graph to the circuit matroid of the original graph. The second polynomial is 
the Bollobas-Riordan polynomial of a ribbon graph, a straightforward 
generalization of the Tutte polynomial. The third polynomial, due V.Krushkal, 
is defined using the symplectic structure in the first homology group of the 
surface.