Ptolemy

The `ptolemy` module

The ptolemy module is part of the popular 3-manifold software SnapPy and can be used to find boundary-unipotent representations of a 3-manifold into pSL(N,C) and computing invariants such as volume and Chern-Simons invariant by creating Ptolemy varieties and finding the solutions to them. The math is explained in:

Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert: Gluing equations for PGL(n,C)-representations of 3-manifolds
Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert: The complex volume of SL(n,C)-representations of 3-manifolds

The ptolemy module automatically either calls Magma or uses sage functions (when used within sage) to compute solutions. We have also precomputed solutions using Magma for many manifolds of the SnapPy census. The results can be downloaded below.

In brief

Type the following lines into sage (with SnapPy installed) or SnapPy (with Magma installed) to compute the volumes and Ptolemy fields of boundary-unipotent representations of the Figure-eight knot complement into PSL(2,C) with non-trivial obstruction class:

from snappy import Manifold M = Manifold("4_1") H = M.ptolemy_obstruction_classes() c = H[1] # non-trivial obstruction class p = M.ptolemy_variety(2, c) sols = p.compute_solutions() [sol.volume_numerical() for sol in sols] [sol.number_field() for sol in sols]

The three new relevant methods of the Manifold class in SnapPy are:

ptolemy_variety
ptolemy_obstruction_classes
gluing_equations_pgl

Each method comes with documentation which can be obtained by starting SnapPy or sage (with SnapPy installed) and typing, e.g.:

from snappy import Manifold
help(Manifold("4_1").ptolemy_variety)

Video tutorial (More coming Shortly)

Introduction - Part 1 - Using withing sage (Watch on youtube)

Data for manifolds of the SnapPy census

We have computed the primary decompositions of the Ptolemy varieties for the manifolds of the SnapPy census using Magma.

Data

SnapPy Orientable Cusped Census
- N=2
  - pSL(2,C) for all census manifolds up to 6 simplices: .tar.bz2 archive (2.3Mb, uncompressed 32Mb)
  - pSL(2,C) for all census manifolds with 7 simplicies: .tar.bz2 archive (38Mb, uncompressed 292Mb)
  - pSL(2,C) for all (but t12063) census manifolds with 8 simplicies: .tar.bz2 archive (863Mb, uncompressed 4.7Gb)
  - pSL(2,C) for t12063: .tar.bz2 archive (154Mb, uncompressed 531Mb)
- N=3
  - SL(3,C) for all (but m019) census manifolds up to 3 simplices: .tar.bz2 archive (10kb, uncompressed 150kb)
  - SL(3,C) for several census manifolds with 4 simplices (coming soon)
- N=4
  - pSL(4,C) for m004 (isometric to 4₁): .tar.bz2 archive (3kb, uncompressed 30kb)
Link complements (coming soon)

Example how to use a file

Download m015__sl3.magma_out (isometric to 5₂) and type into SnapPy or sage

from snappy import ptolemy cd /PATH_TO_WHERE_FILE_IS sols = ptolemy.solutions_from_magma_file("m015__sl3.magma_out") [sol.volume_numerical() for sol in sols]