Open Access
January 2014 The skein algebra of arcs and links and the decorated Teichmüller space
Julien Roger, Tian Yang
J. Differential Geom. 96(1): 95-140 (January 2014). DOI: 10.4310/jdg/1391192694


We define an associative $\mathbb{C}[[h]]$-algebra $\mathcal{A}\mathcal{S}_h(\Sigma)$ generated by regular isotopy classes of arcs and links over a punctured surface $\Sigma$ which is a deformation quantization of the Poisson algebra $\mathcal{C}(\Sigma)$ of arcs and loops on $\Sigma$ endowed with a generalization of the Goldman bracket. We then construct a Poisson algebra homomorphism from $\mathcal{C}(\Sigma)$ to the algebra of smooth functions on the decorated Teichmüller space endowed with a natural extension of the Weil-Petersson Poisson structure described by Mondello. 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.


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Julien Roger. Tian Yang. "The skein algebra of arcs and links and the decorated Teichmüller space." J. Differential Geom. 96 (1) 95 - 140, January 2014.


Published: January 2014
First available in Project Euclid: 31 January 2014

zbMATH: 1290.53080
MathSciNet: MR3161387
Digital Object Identifier: 10.4310/jdg/1391192694

Rights: Copyright © 2014 Lehigh University

Vol.96 • No. 1 • January 2014
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