Ribbon graphs and the Temperley-Lieb Algebra

TL;DR

This paper constructs a new algebra based on ribbon graphs and skein relations, demonstrating its relation to the Temperley-Lieb algebra and providing explicit algebraic structures and generators.

Contribution

It introduces the algebra ${\\mathcal Y}_n$ using ribbon graphs and skein relations, establishing its connection to the Temperley-Lieb algebra and explicitly describing ${\mathcal Y}_2$ and ${\mathcal Y}_3$.

Findings

${\mathcal Y}_2$ is isomorphic to a quotient of a three-variable polynomial algebra

Provided a family of generators for ${\mathcal Y}_3$

Established the relation between ribbon graph-based algebra and Temperley-Lieb algebra

Abstract

Let $n$ be a nonnegative integer, we use ribbon $n-$graph diagrams and the Yamada polynomial skein relations to construct an algebra ${\mathcal Y}_n$ which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra ${\mathcal Y}_2$ is isomorphic to some quotient of a three variables polynomial algebra. Then, we give a family of generators for the algebra ${\mathcal Y}_3$.