Graph polynomials and link invariants as positive type functions on Thompson's group F

In a recent paper, Jones introduced a correspondence between elements of the Thompson group F and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of F. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of N-colorings and the Tutte polynomial, can be viewed as positive definite functions on F.