$A_2$ Skein Representations of Pure Braid Groups

Abstract

We define a family of representations $\{{\rho }_n{\}}_{n\ge 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its boundary. We also introduce a triangle-free basis of such an $A_2$ web space and calculate matrix representations of ${\rho }_n$ about the standard generators of $P_{2k}$.

13 pages, many TikZ pictures