Ricerc@Sapienza

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups F→ and T→ of the Thompson groups F and T, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones–Thompson subgroups.

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.