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

TL;DR

This paper demonstrates that certain link invariants and graph polynomials, including the Kauffman bracket, can be viewed as positive definite functions on Thompson's group F, using elementary and statistical mechanics methods.

Contribution

It provides new elementary proofs and interpretations of link invariants as positive definite functions on F, expanding the connection between graph polynomials and group representations.

Findings

Kauffman bracket at roots of unity is a positive definite function on F.

Number of N-colorings and Tutte polynomial are positive definite functions on F.

Elementary proof techniques via statistical mechanics models.

Abstract

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 $F$.