The skein algebra of arcs and links and the decorated Teichmuller space

2011/10/12 by Julien Roger, Tian Yang, Roger, Julien +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1110.2748

Abstract

This dissertation is based on a joint work with Dr. Julien Roger. We define an associative mathbbC[[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.

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