#### Volume 15, issue 1 (2015)

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

### Stavros Garoufalidis, Matthias Goerner and Christian K Zickert

Algebraic & Geometric Topology 15 (2015) 565–622
##### Abstract

Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3–manifold group into $SL\left(n,ℂ\right)$ using Ptolemy coordinates, which were inspired by $\mathsc{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We parametrize representations into $PGL\left(n,ℂ\right)$ using shape coordinates, which are a $3$–dimensional analogue of Fock and Goncharov’s $\mathsc{X}$–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.

##### Keywords
generalized gluing equations, shape coordinates, Ptolemy coordinates, Neumann–Zagier datum
##### Mathematical Subject Classification 2010
Primary: 57M27, 57N10
Secondary: 53D50