DescriptionThis dissertation is based on a joint work with Dr. Julien Roger. We define an associative $mathbb{C}[[h]]$--algebra $AS_h(Sigma)$ 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.