Constructing Thompson representatives via pointed links

TL;DR

This paper extends Jones' construction to map the Brown-Thompson group $F_3$ onto pointed links, introduces a new monoid structure, and establishes a standard form for link representatives using algebraic operations.

Contribution

It develops a novel algebraic framework connecting $F_3$ to pointed links, including a new monoid and standard forms for link representatives.

Findings

Surjective map from $F_3$ to pointed links established.

Introduction of the central monoid $(F_3, riangle)$.

Standard form for connected sum representatives in $F_3$.

Abstract

We extend Jones' construction to obtain a surjective map from the Brown-Thompson group $F_3$ to the set of pointed links up to pointed isotopy. We then introduce an operation on $F_3$, and use it to define a new monoid $(F_3, \diamond)$, called the central monoid. Using the extended version of Jones' construction, we obtain a surjective monoid homomorphism from the central monoid to the monoid of pointed links with connected sum. This allows us to introduce a standard form for connected sum representatives in $F_3$, and we extend this construction to a certain family of links by defining disjoint union and linking moves on $F_3$.