Explicit representations and Azumaya loci of skein algebras of small surfaces

Abstract

We construct finite dimensional representations of the Kauffman bracket skein algebra of the one-punctured torus and four-punctured sphere at all roots of unity. The representations are given by explicit formulas. They all have dimensions equal to the PI degrees of the skein algebras, and they realize all classical shadows. We then use the reducibility of these representations to determine the Azumaya loci. In particular, the Azumaya loci of these surfaces contain the smooth loci of the classical shadow varieties, with equality in the case of the one-punctured torus and proper containment in the case of the four-punctured sphere.