Regular isotopy classes of link diagrams from Thompson’s groups

Abstract

In 2014, Vaughan Jones developed a method to produce links from elements of Thompson’s group [Formula: see text], and showed that all links arise this way. He also introduced a subgroup [Formula: see text] of [Formula: see text] 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 [Formula: see text], as well as exactly those regular isotopy classes of oriented links that arise from [Formula: see text], answering a question asked by Jones in 2018.