Representations of the Kauffamn skein algebra of small surfaces

TL;DR

This paper establishes a uniqueness theorem for finite-dimensional irreducible representations of the Kauffman skein algebra of small surfaces at roots of unity, linking classical invariants to representation isomorphism.

Contribution

It proves a new uniqueness result for irreducible representations of the skein algebra of small surfaces, under generic classical shadow conditions.

Findings

Unique correspondence between classical shadows and representations

Isomorphism of representations with same invariants

Applicability to spheres with ≤4 punctures and tori with ≤1 puncture

Abstract

We prove a uniqueness result for finite-dimensional representations of the Kauffman skein algebra $\mathcal{S}_A(S)$ of a surface $S$, when $A$ is a root of unity and when the surface $S$ is a sphere with at most four punctures or a torus with at most one puncture. We show that, if two irreducible representations of $\mathcal{S}_A(S)$ have the same classical shadow and the same puncture invariants, and if this classical shadow is sufficiently generic in the character variety $\mathcal{X}_{SL_2(\mathbb{C})} (S)$, then the two representations are isomorphic.