Framization of the Temperley-Lieb Algebra

TL;DR

This paper introduces a new algebraic structure called the framization of the Temperley-Lieb algebra, extending it via the Yokonuma-Hecke algebra to create novel knot invariants that differ from the Jones polynomial.

Contribution

It defines the framization as a quotient of the Yokonuma-Hecke algebra and establishes conditions for a Markov trace to produce new knot invariants.

Findings

Constructed new knot invariants from the framized algebra.

Proved these invariants are not equivalent to the Jones polynomial.

Provided algebraic conditions for the trace to pass to the quotient.

Abstract

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke 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. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.