Gluing equations for PGL(n, ℂ)–representations of 3–manifolds

By Stavros Garoufalidis, Matthias Goerner, C. Zickert. 2012

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Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3‐manifold group into SL.n;C/ using Ptolemy coordinates, which were inspired by A‐coordinates on higher Teichmuller space due to Fock and Goncharov. We parametrize representations into PGL.n;C/ 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b11cd7fa09f8ddbeb1720c86f9064d618fd54c1