1110.2748 (Julien Roger et al.)

The skein algebra of arcs and links and the decorated Teichmüller
space    [PDF]

Julien Roger, Tian Yang

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.

View original: http://arxiv.org/abs/1110.2748