  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 $N$-colourings and the Tutte
polynomial, can be viewed as positive definite functions on $F$.
