Conway Rational Tangles and the Thompson Group

TL;DR

This paper explores the connection between Conway rational tangles and the Thompson group, providing methods to construct knots and links via Thompson group elements using tangle concatenation.

Contribution

It introduces a new approach to constructing knots from Thompson group elements through Conway rational tangles, advancing skein-theoretic methods.

Findings

Methods to construct any product of simple tangles

Framework for generating knots via Thompson group elements

Potential for skein-theoretic approaches to knot construction

Abstract

There is a map, defined and studied by Jones, from Thompson's group $F$ to knots. Jones proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element. We approach the question of methodologically finding Thompson group elements to generate a particular knot or link through the lens of Conway's rational tangles. We are able to give methods to construct any product or concatenation of simple tangles, and we hope these are seeds for a more skein-theoretic approach to the construction question.