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

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 $T$.

To appear in Complex Analysis and Operator Theory

References in corpus (2)

• A no-go theorem for the continuum limit of a periodic quantum spin chain
• The Homflypt polynomial and the oriented Thompson group

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