Regular Isotopy Classes of Link Diagrams From Thompson's Groups

TL;DR

This paper classifies the specific regular isotopy classes of links and oriented links that can be generated from Thompson's group $F$ and its subgroup $ar{F}$, answering a question posed by Jones.

Contribution

It provides a complete classification of link classes arising from Thompson's groups, clarifying the scope of links generated by these algebraic structures.

Findings

Classified regular isotopy classes of links from $F$

Classified regular isotopy classes of oriented links from $ar{F}$

Answered Jones's 2018 question about link generation from Thompson's groups

Abstract

In 2014, Vaughan Jones developed a method to produce links from elements of Thompson's group $F$, and showed that all links arise this way. He also introduced a subgroup $\vec{F}$ of $F$ and a method to produce oriented links from elements of this subgroup. In 2018, Valeriano Aiello showed that all oriented links arise from this construction. We classify exactly those regular isotopy classes of links that arise from $F$, as well as exactly those regular isotopy classes of oriented links that arise from $\vec{F}$, answering a question asked by Jones in 2018.