The skein algebra of arcs and links and the decorated Teichmüller space

Details The skein algebra of arcs and links and the decorated Teichmüller space The skein algebra of arcs and links and the decorated Teichmüller space

2011-10-12

arxiv.org/abs/1110.2748v2

Mathematics

Geometric Topology

39 pages, many figures

Scientific paper

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with the Weil-Petersson Poisson structure. The construction relies on a collection of geodesic lengths identities in hyperbolic geometry which generalize Penner's Ptolemy relation, the trace identities and Wolpert's cosine formula. As a consequence, we derive an explicit formula for the geodesic lengths functions in terms of the edge lengths of an ideally triangulated decorated hyperbolic surface.

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