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.

A generalized skein relation for Khovanov homology and a categorification of the $θ$-invariant

“…For the rest of the paper, we will only be interested in this case, that is, when E is the inverse of a positive integer d. For d = 1, the invariants Θ and θ coincide with the HOMFLYPT and the Jones polynomial respectively. In general, by [6,Theorem 8.2] and [8, Theorem 5] (see also [5,Example 4.16]), we have: Theorem 3. The invariants Θ and θ are stronger than the HOMFLYPT and the Jones polynomial respectively.…”

“…Therefore, as we also explain in Section 3.2, the computation of θ can be seen as a dynamical system whose initial condition is the value of θ on a union of unlinked knots: if L is union of r unlinked knots, we have θ(L) = d r−1 J(L). The invariant θ is stronger than the Jones polynomial, in the sense that it distinguishes links that the Jones polynomial cannot distinguish [6,8,5].…”

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

“…The invariant θ is stronger than the Jones polynomial, in the sense that it distinguishes links that the Jones polynomial cannot distinguish [5,6,8].…”