The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups $\vec{F}$ and $\vec{T}$

TL;DR

This paper explores the relationship between knot invariants like the Jones polynomial and functions of positive type on Thompson groups, introducing new interpretations and extending results to other link invariants such as the Kauffman and HOMFLY polynomials.

Contribution

It provides an alternative partition function approach to link invariants and demonstrates their connection to positive type functions on Thompson groups and their subgroups.

Findings

Evaluations of the Jones polynomial relate to unitary representations of Thompson groups.

Extensions to the Kauffman and HOMFLY polynomials as functions of positive type.

New methods linking knot invariants with algebraic structures of Thompson groups.

Abstract

The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ 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. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups $\vec{F}$ and $\vec{T}$. Within this framework, we discuss an alternative approach that relies on some partition function interpretation of the Jones polynomial, and also exhibit more examples associated with other link invariants, notably the two-variable Kauffman polynomial and the HOMFLY polynomial. In the unoriented case, extending our previous results, we also show by similar methods that certain evaluations of the Tutte polynomial and of the Kauffman bracket, suitably renormalized, yield functions of positive type on $T$.