| This dissertation is based on a joint work with Dr. Julien Roger. We define an associative C [[h]]--algebra ASh (Sigma) generated by framed arcs and links over a punctured surface Sigma which is a quantization of the Poisson algebra C (Sigma) of arcs and curves on Sigma. We also construct a Poisson algebra homomorphism from C (Sigma) 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 generalizes Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula. |
