  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.
