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 PGL(N,C) and compute invariants such as volume and Chern-Simons invariant. The Ptolemy varities can be computed using Magma or Sage or retrieved automatically from our precomputed database.

Data for SnapPy manifolds


Browsable database

Individual magma files

Tar archives

Our data contains the primary decompositions of the Ptolemy varieties as well as the volume of each representation (see notes and caveats for more). The manifolds for which we have complete data include:
  • PGL(2,C)
    • OrientableCuspedCensus: all manifolds up to 9 simplices
    • LinkExteriors: all manifolds up to 16 simplices
    • HTLinkExteriors: all manifolds up to 13 simplices but L14n31302
  • PGL(3,C)
    • OrientableCuspedCensus: all manifolds up to 3 simplices and selected manifolds with 4
  • PGL(4,C)
    • m004
New: We also list the trace fields (of the geometric representation) of all OrientableCuspedCensus manifolds.


Please report bugs to enischte at gmail dot com.