Journal of Differential Geometry

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

Julien Roger and Tian Yang

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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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J. Differential Geom., Volume 96, Number 1 (2014), 95-140.

First available in Project Euclid: 31 January 2014

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Roger, Julien; Yang, Tian. The skein algebra of arcs and links and the decorated Teichmüller space. J. Differential Geom. 96 (2014), no. 1, 95--140. doi:10.4310/jdg/1391192694.

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