We show how to construct unitary representations of the oriented Thompson group F from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of F.
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.
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