From the Framisation of the Temperley–Lieb Algebra to the Jones Polynomial: An Algebraic Approach

Abstract

We prove that the Framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. We use this result to obtain a closed combinatorial formula for the invariants for classical links obtained from a Markov trace on the Framisation of the Temperley–Lieb algebra. For a given link L, this formula involves the Jones polynomials of all sublinks of L, as well as linking numbers.