Algebraic & Geometric Topology

Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds

Stavros Garoufalidis, Matthias Goerner, and Christian Zickert

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Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3–manifold group into SL(n, ) using Ptolemy coordinates, which were inspired by A–coordinates on higher Teichmüller space due to Fock and Goncharov. We parametrize representations into PGL(n, ) using shape coordinates, which are a 3–dimensional analogue of Fock and Goncharov’s X–coordinates. These coordinates satisfy equations generalizing Thurston’s gluing equations. These equations are of Neumann–Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.

Article information

Algebr. Geom. Topol., Volume 15, Number 1 (2015), 565-622.

Received: 7 November 2014
Accepted: 13 December 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 53D50: Geometric quantization

generalized gluing equations shape coordinates Ptolemy coordinates Neumann–Zagier datum


Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds. Algebr. Geom. Topol. 15 (2015), no. 1, 565--622. doi:10.2140/agt.2015.15.565.

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