The Skein Algebra and the Decorated Teichmuller Space

Tian Yang (Rutgers)

Abstract: The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytycki and Turaev as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock-Frohman-Kania-Bartoszynska and Przytycki-Sikora that K(S X [0,1]) is a quantization of the SL(2,C) characters of the fundamental group of $S$ with respect to the Goldman-Weil-Petersson Poisson bracket. In a joint work with J. Roger, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in SX [0,1] satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on S in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmuller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.