The Jones polynomial of an alternating link is a certain specialization of
the Tutte polynomial of the (planar) checkerboard graph associated to an
alternating projection of the link. The Bollobas-Riordan-Tutte polynomial
generalizes the Tutte polynomial of planar graphs to graphs that are embedded
in closed oriented surfaces of higher genus.
In this paper we show that the Jones polynomial of any link can be obtained
from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph
associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change