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

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 $F$.

To appear in Journal of Knot Theory and Its Ramifications

References in corpus (4)

• A no-go theorem for the continuum limit of a periodic quantum spin chain
• The Homflypt polynomial and the oriented Thompson group
• A computational approach to the Thompson group $F$
• The Jones polynomial and functions of positive type on the oriented Jones-Thompson groups $\vec{F}$ and $\vec{T}$

Cited by in corpus (8)

• On the Alexander Theorem for the oriented Thompson group $\vec{F}$
• Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebras
• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• On the oriented Thompson subgroup ${\vec{F}}_3$ and its relatives in higher Brown-Thompson groups
• Positive oriented Thompson links
• An introduction to Thompson knot theory and to Jones subgroups
• The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part
• Remarks on some maximal subgroups of $F$ and on the $\vec{F}$-index of knots