Volume 15, issue 1 (2015)

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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 $SL\left(n,ℂ\right)$–representations of a $3$–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the $\mathsc{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) $PSL\left(2,ℂ\right)$-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
Primary: 57N10
Secondary: 57M27