A Survey on Temperley-Lieb-type quotients from the Yokonuma-Hecke algebras

TL;DR

This survey reviews the construction of Temperley-Lieb-type quotients from Yokonuma-Hecke algebras, exploring their properties, link invariants, and potential for more powerful link distinguishing invariants.

Contribution

It systematically compiles results on quotient algebras, their representation theory, and introduces new link invariants surpassing the Jones polynomial.

Findings

Identified conditions for Markov trace factorization.

Derived new link invariants distinguishing more links.

Proposed a stronger two-variable invariant for classical links.

Abstract

In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type $A$ as a quotient algebra of the Yokonuma-Hecke algebra of type $A$. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimension, linear bases, representation theory and the necessary and sufficient conditions for the unique Markov trace of the Yokonuma-Hecke algebra to factor through to each one of them. Further, we present the link invariants that are derived from each quotient algebra and we point out which quotient algebra provides the most natural definition for a framization of the Temperley-Lieb algebra. From the Framization of the Temperley-Lieb algebra we obtain new one-variable invariants for oriented classical links that, when compared to the Jones polynomial, they are not topologically equivalent since they distinguish more pair of non isotopic oriented links. Finally, we discuss the generalization of the newly obtained invariants to a new two-variable invariant for oriented classical links that is stronger than the Jones polynomial.