A cellular basis for the generalized Temperly-Lieb Algebra and Mahler Measure

TL;DR

This paper establishes a cellular algebra structure for a colored skein algebra related to the Temperley-Lieb algebra, introduces Jucys-Murphy elements, and connects these to Mahler measure convergence of Jones polynomials.

Contribution

It provides an explicit cellular basis and Jucys-Murphy elements for the colored skein algebra, enabling new insights into the algebra's structure and Mahler measure relations.

Findings

Cellular basis for the colored skein algebra is constructed.

Jucys-Murphy elements are explicitly identified.

Mahler measure of Jones polynomial converges under twisting.

Abstract

Just as the Temperley-Lieb algebra is a good place to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with $2k$ colored points on the boundary, each with color $n$, is a good place to compute the $n^{th}$ colored Jones polynomial. Here, this colored skein algebra is shown to be a cellular algebra and a set of separating Jucys-Murphy elements is provided. This is done by explicitly providing the cellular basis and the JM-elements. Having done this several results of Mathas on such algebras are considered, including the construction of pairwise non-isomorphic irreducible submodules and their corresponding primitive idempotents. These idempotents are then used to define recursive elements of the colored skein algebra. Recursive elements are of particular interest as they have been used to relate geometric properties of link diagrams to the Mahler measure of the Jones polynomial. In particular, a single proof is given for the result of Champanerkar and Kofman, that the Mahler measure of the Jones and colored Jones polynomial converges under twisting on some number of strands.