$A_2$ Skein Representations of Pure Braid Groups

TL;DR

This paper introduces a new family of representations of pure braid groups derived from their action on specialized $A_2$ web spaces, generalizing skein modules and providing explicit matrix forms.

Contribution

It defines $ ho_n$ representations from $P_{2k}$ acting on $A_2$ web spaces, introduces a triangle-free basis, and computes explicit matrix representations for standard generators.

Findings

Defined $ ho_n$ representations of pure braid groups.

Introduced a triangle-free basis for $A_2$ web spaces.

Calculated explicit matrix forms of the representations.

Abstract

We define a family of representations $\{\rho_n\}_{n\geq 0}$ 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 $P_{2k}$.