Alexander's theorem for stabilizer subgroups of Thompson's group

TL;DR

This paper proves that most stabilizer subgroups of Thompson's group F, under its natural action on the interval, satisfy Alexander's theorem, extending the understanding of knot constructions from these subgroups.

Contribution

It establishes that nearly all stabilizer subgroups of Thompson's group F satisfy Alexander's theorem, revealing new insights into their structure and knot representation capabilities.

Findings

Most stabilizer subgroups satisfy Alexander's theorem

Thompson's group F has diverse subgroups with varying properties

Extension of Alexander's theorem to a broad class of subgroups

Abstract

In 2017, Jones studied the unitary representations of Thompson's group $F$ and defined a method to construct knots and links from $F$. One of his results is that any knot or link can be obtained from an element of this group, which is called Alexander's theorem. On the other hand, Thompson's group $F$ has many subgroups and it is known that there exist various subgroups which satisfy or do not satisfy Alexander's theorem. In this paper, we prove that almost all stabilizer subgroups under the natural action on the unit interval satisfy Alexander's theorem.