Journal of Differential Geometry

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

Julien Roger and Tian Yang

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1391192694

Digital Object Identifier
doi:10.4310/jdg/1391192694

Mathematical Reviews number (MathSciNet)
MR3161387

Zentralblatt MATH identifier
1290.53080

Citation

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. https://projecteuclid.org/euclid.jdg/1391192694


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