Geodesic
The Ptolemy field of 3–manifold representations
Garoufalidis, Stavros1 ; Goerner, Matthias2 ; Zickert, Christian3
1School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
2Pixar Animation Studios, 1200 Park Avenue, Emeryville, CA 94608, USA
3Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States
Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 371-397
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The Ptolemy coordinates for boundary-unipotent SL(n, ℂ)–representations of a 3–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the A–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) PSL(2, ℂ)-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

DOI : 10.2140/agt.2015.15.371
Classification : 57N10, 57M27
Keywords: Ptolemy coordinates, trace field, SnapPy, $3$–manifold

Garoufalidis, Stavros  1   ; Goerner, Matthias  2   ; Zickert, Christian  3

1 School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
2 Pixar Animation Studios, 1200 Park Avenue, Emeryville, CA 94608, USA
3 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States 
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Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. The Ptolemy field of 3–manifold representations. Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 371-397. doi: 10.2140/agt.2015.15.371

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