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 SL 2 (C)-characters of the fundamental group of F , corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined. Topologically free Kauffman bracket modules are shown to generate finite type invariants. It is shown 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 finitely generated.

We construct lattice gauge field theory based on a quantum group on a lattice of dimension 1. Innovations include a coalgebra structure on the connections, and an investigation of connections that are not distinguishable by observables. We prove that when the quantum group is a deformation of a connected algebraic group (over the complex numbers), then the algebra of observables forms a deformation quantization of the ring of characters of the fundamental group of the lattice with respect to the corresponding algebraic group. Finally, we investigate lattice gauge field theory based on quantum SL(2,C), and conclude that the algebra of observables is the Kauffman bracket skein module of a cylinder over a surface associated to the lattice.Comment: 35 pages, amslatex, epsfig, many figures; email addresses: bullock@math.gwu.edu, frohman@math.uiowa.edu, kania@diamond.idbsu.ed

Let F be a finite type surface and ζ a complex root of unity. The Kauffman bracket skein algebra K ζ (F ) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of K ζ (F ) over its center, and we extend a theorem of [8] which says the skein algebra has a splitting coming from two pants decompositions of F .

Abstract. For each closed, orientable surface Σg, we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module Kt(Σg × I). The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = −1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of Σg.

The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The terms of the sum can be indexed by surfaces lying in the three-manifold. The contribution of a surface is largest when the surface is normal and when its genus is the lowest.1991 Mathematics Subject Classification. 57M27. Key words and phrases. Turaev-Viro invariants of 3-manifolds, normal surface, regular spine, quantum 6j-symbol.