The Yokonuma-Temperley-Lieb Algebra

TL;DR

This paper introduces the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra and establishes conditions under which it yields knot invariants equivalent to the Jones polynomial.

Contribution

It defines a new algebraic structure and connects it to knot invariants, extending the classical Temperley-Lieb algebra framework.

Findings

Yokonuma-Temperley-Lieb algebra is a quotient of Yokonuma-Hecke algebra.

Necessary and sufficient conditions for the Markov trace to pass to the quotient.

Sequence of knot invariants coincides with the Jones polynomial.

Abstract

In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra, leading to a sequence of knot invariants which coincide with the Jones polynomial.