|
Organizers |
The Skein Algebra of Punctured Surfaces
by
Tian Yang
Mathematics Department, Rutgers University-New Brunswick.
Coauthors: Julien Roger
The Kauffman bracket skein module K(M) of a 3-manifold M is defined by Przytycki as an invariant for framed links in M satisfying the Kauffman skein relation. For a compact oriented surface S, it is shown by Bullock, Przytycki, Sikora, Frohman and Kania-Bartoszynska that K(S×I) is a quantization of the SL2C-characters of the fundamental group of S in the Goldman-Weil-Petersson Poisson structure.
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 S×I 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 Teichmüller 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.
Date received: October 3, 2011
Copyright © 2011 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbdt-03.