The Ptolemy field of 3–manifold representations

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.

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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

Volume 15, issue 1 (2015)

Stavros Garoufalidis, Matthias Goerner and Christian K Zickert

Abstract