Representations of the Kauffamn skein algebra of small surfaces

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.