The skein algebra of arcs and links and the decorated Teichm\"uller space

TL;DR

This paper introduces a quantized algebra of arcs and links on punctured surfaces, connecting it to the decorated Teichmüller space via Poisson algebra homomorphisms and hyperbolic geometry identities.

Contribution

It constructs a new associative algebra quantization of the Poisson algebra of arcs and links, linking hyperbolic geometry identities to the structure of the decorated Teichmüller space.

Findings

Defined the algebra AS_h(S) for punctured surfaces.

Constructed a Poisson algebra homomorphism to the decorated Teichmüller space.

Derived explicit formulas for geodesic length functions.

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.