Multiplicative structure of kauffman bracket skein module quantizations

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(so₃)). For a torus without boundary we obtain a quantization of "the symmetric homologies" of a torus (equivalently, the coordinate ring of the SL₂(ℂ)-character variety of ℤ ⊕ 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.