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The Ptolemy field of $3$–manifold representations

Stavros Garoufalidis, Matthias Goerner and Christian K Zickert

Algebraic & Geometric Topology 15 (2015) 371–397
Abstract

The Ptolemy coordinates for boundary-unipotent SL(n, )–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, )-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.

Keywords
Ptolemy coordinates, trace field, SnapPy, $3$–manifold
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57M27
References
Publication
Received: 21 January 2014
Revised: 9 May 2014
Accepted: 7 July 2014
Published: 23 March 2015
Authors
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
USA
http://www.math.gatech.edu/~stavros
Matthias Goerner
Pixar Animation Studios
1200 Park Avenue
Emeryville, CA 94608
USA
http://www.unhyperbolic.org/
Christian K Zickert
Department of Mathematics
University of Maryland
College Park, MD 20742-4015
United States
http://www2.math.umd.edu/~zickert