The Jones polynomial and dessins d'enfant

preprint available here

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 surfaces of
higher genus (i.e. dessins d'enfant).
In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte
polynomial of a certain dessin associated to a link projection. We give some applications of this approach.

Jones Polynomial, Dessins d'enfant, Bollobas-Riordan-Tutte polynomial, spanning tree expansion