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. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups F→ and 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.

The Jones polynomial and functions of positive type on the oriented Jones–Thompson groups F→ and T→ / Aiello, V.; Conti, R.. - In: COMPLEX ANALYSIS AND OPERATOR THEORY. - ISSN 1661-8254. - 13:7(2019), pp. 3127-3149. [10.1007/s11785-018-0866-6]

The Jones polynomial and functions of positive type on the oriented Jones–Thompson groups F→ and T→

Aiello V.;Conti R.
2019

Abstract

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. One important example is provided by suitable evaluations of the Jones polynomial, which are thus associated with certain unitary representations of the groups F→ and 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.
2019
2-variable Kauffman polynomial; Chromatic polynomial; Fox colourings; Function of positive type; HOMFLY polynomial; Jones polynomial; Kauffman bracket; Knots; Link colourings; Links; Partition function; Rank polynomial; Thompson group; Trees; Tutte polynomial
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The Jones polynomial and functions of positive type on the oriented Jones–Thompson groups F→ and T→ / Aiello, V.; Conti, R.. - In: COMPLEX ANALYSIS AND OPERATOR THEORY. - ISSN 1661-8254. - 13:7(2019), pp. 3127-3149. [10.1007/s11785-018-0866-6]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1324597
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