A computational study of the number of connected components of positive Thompson links

TL;DR

This paper introduces Thompson permutations derived from Thompson groups to analyze the number of connected components in positive Thompson links, supported by numerical experiments and theoretical proofs.

Contribution

It defines Thompson permutations for elements of Thompson groups and proves all oriented links can be generated through this method.

Findings

Number of orbits matches the number of link components

Exploration of positive elements of F_3 with fixed width and height

Numerical conjectures on link components based on Thompson permutations

Abstract

Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups $F$, which was later extended to the Brown-Thompson group $F_3$. In this article we define a way to produce permutations out of elements of the $F$ and $F_3$ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of $F_3$ of fixed \emph{width} and \emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of $F$ and $F_3$. We prove that all oriented links can be produced in this way.