Some unitary representations of Thompson's groups F and T

TL;DR

This paper constructs a family of unitary representations of Thompson's groups F and T derived from subfactors, linking them to polynomial invariants of links and introducing new subgroups to generate all oriented knots and links.

Contribution

It introduces a novel method to obtain unitary representations of Thompson's groups related to algebraic quantum field theories and link invariants, expanding the understanding of their algebraic and topological properties.

Findings

Representations correspond to polynomial link invariants.

All links and oriented knots can be generated within this framework.

New oriented subgroups enable comprehensive knot and link construction.

Abstract

In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the "local scale transformations" of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new "oriented" subgroups $\overrightarrow F <F$ and $\overrightarrow T< T$ which allow us to produce all \emph{oriented} knots and links.