On Jones' subgroup of R. Thompson group $F$

TL;DR

This paper explores the subgroup  of R. Thompson group F, showing it is generated by specific elements, isomorphic to F_3, and coincides with its commensurator, with implications for its linear representations.

Contribution

It proves that  is generated by three elements, isomorphic to F_3, and equals its own commensurator, answering several of Jones's questions.

Findings

 is generated by x_0x_1, x_1x_2, x_2x_3

 is isomorphic to F_3

 coincides with its commensurator

Abstract

Recently Vaughan Jones showed that the R. Thompson group $F$ encodes in a natural way all knots, and a certain subgroup $\vec F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\vec F$. In particular we prove that the subgroup $\vec F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i, i=0,1,2,...$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\vec F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\vec F$ is irreducible.