Half grid diagrams and Thompson links

TL;DR

This paper introduces half grid diagrams, proves every link can be represented by them, and relates these diagrams to Thompson groups and classical link invariants, providing new insights into link theory.

Contribution

It defines half grid diagrams, establishes their equivalence with Jones' Thompson link construction, and connects the Thompson index to classical topological invariants.

Findings

Every link is half grid presentable.

Established the equivalence between half grid diagrams and Thompson links.

Bounded the Thurston-Bennequin number using the Thompson index.

Abstract

We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We show that this half grid construction is equivalent to Jones' construction of oriented Thompson links. Using this equivalence, we relate the (oriented) Thompson index to several classical topological link invariants, and give both the lower and upper bounds of the maximal Thurston-Bennequin number of a knot in terms of the oriented Thompson index. Moreover, we give a one-to-one correspondence between half grid diagrams and elements in symmetric groups and give a new description of link group using two elements in a symmetric group.