Framization of a Temperley-Lieb algebra of type $\mathtt{B}$

TL;DR

This paper extends the Framization of the Temperley-Lieb algebra to Coxeter systems of type B, defining a new algebra with a unique Markov trace that leads to invariants for framed and classical links in the solid torus.

Contribution

It introduces the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra, with conditions for the Markov trace to pass, enabling link invariants.

Findings

Defined a natural extension of Temperley-Lieb algebra to type B

Proved the existence of a unique Markov trace on this extension

Constructed invariants for links in the solid torus

Abstract

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type $\mathtt{B}$ as a quotient of the Yokonuma-Hecke algebra of type $\mathtt{B}$. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type $\mathtt{B}$ to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.