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

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.