Multiplicative structure of Kauffman bracket skein module quantizations

Abstract:

We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra closely related to a cyclic quantization of $U(\mathfrak {so}_3$)). For a torus without boundary we obtain a quantization of “the symmetric homologies" of a torus (equivalently, the coordinate ring of the $SL_2(\mathbb {C})$-character variety of $\mathbb {Z}\oplus \mathbb {Z}$). Presentations are also given for the four-punctured sphere and twice-punctured torus. We conclude with an investigation of central elements and zero divisors.