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

Julien Roger; Tian Yang

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arXiv · 1110.2748

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

Julien Roger·Tian Yang

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Abstract

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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BibTeX RIS

Julien Roger, Tian Yang. 2012-02-20. The skein algebra of arcs and links and the decorated Teichmüller space. https://arxiv.org/abs/1110.2748

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