A cellular basis for the generalized Temperley–Lieb algebra and Mahler measure

Abstract

Just as the Temperley–Lieb algebra is a good tool 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 tool to compute the nth colored Jones polynomial. We show that the colored skein algebra is a cellular algebra and find a set of separating Jucys–Murphy elements. 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 any number of strands.