The $3$-colorable subgroup of Thompson's group and tricolorability of links

TL;DR

This paper demonstrates that all elements of the 3-colorable subgroup of Thompson's group $F$ produce 3-colorable links, connecting group theory with knot theory and link colorability.

Contribution

It establishes a direct link between the 3-colorable subgroup of Thompson's group and the 3-colorability of links derived from its elements, a novel connection in the field.

Findings

All elements in the 3-colorable subgroup produce 3-colorable links.

The work bridges group theory and knot theory through link colorability.

Provides a new perspective on the structure of Thompson's group and link invariants.

Abstract

Starting from the work by Jones on representations of Thompson's group $F$, subgroups of $F$ with interesting properties have been defined and studied. One of these subgroups is called the $3$-colorable subgroup $\mathcal{F}$, which consists of elements whose ``regions'' given by their tree diagrams are $3$-colorable. On the other hand, in his work on representations, Jones also gave a method to construct knots and links from elements of $F$. Therefore it is a natural question to explore a relationship between elements in $\mathcal{F}$ and $3$-colorable links in the sense of knot theory. In this paper, we show that all elements in $\mathcal{F}$ give 3-colorable links.