The $p$-colorable subgroup of Thompson's group

TL;DR

This paper generalizes the concept of the 3-colorable subgroup of Thompson's group to p-colorable subgroups for odd integers p, showing these subgroups produce p-colorable knots and links and are isomorphic to certain Brown–Thompson groups.

Contribution

It introduces p-colorable subgroups of Thompson's group for odd p, extending previous work on 3-colorability, and proves their isomorphism to specific Brown–Thompson groups.

Findings

Non-trivial elements of the p-colorable subgroup produce p-colorable knots and links.

The p-colorable subgroup is isomorphic to a certain Brown–Thompson group.

Generalization from 3-colorable to p-colorable subgroups for odd p.

Abstract

Recently, Jones introduced a method of constructing knots and links from elements of Thompson's group $F$ by using its unitary representations. He also defined several subgroups of $F$ as the stabilizer subgroups and some researchers studied them algebraically. One of the subgroups is called the 3-colorable subgroup $\mathcal{F}$, and the authors proved that all knots and links obtained from non-trivial elements of $\mathcal{F}$ are 3-colorable. In this paper, for any odd integer $p$ greater than two, we define the $p$-colorable subgroup of $F$ whose non-trivial elements yield $p$-colorable knots and links and show it is isomorphic to the certain Brown--Thompson group.