Understanding the Kauffman Bracket Skein Module

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = e h/4 . For a compact oriented surface F, it is shown that K(F×I) is a quantization of the [Formula: see text]-characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine [Formula: see text]-characters of π 1 (M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.