The Kauffman bracket skein module $K(M)$ of a $3$-manifold $M$ is defined by Przytycki and Turaev as an invariant for framed links in $M$ satisfying the Kauffman skein relation. For a compact oriented surface $S$, it is shown by Bullock--Frohman--Kania-Bartoszynska and Przytycki-Sikora that $K(S\times [0,1])$ is a quantization of the $SL_2\mathbb{C}$-characters of the fundamental group of $S$ with respect to the Goldman--Weil--Petersson Poisson bracket.
In a joint work with J. Roger, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in $S\times [0,1]$ satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on $S$ in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmuller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.