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The Skein Algebra and the Decorated Teichmuller Space

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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.