The ptolemy module

What is the ptolemy module?

This module provides tools to find boundary-unipotent representations of an oriented 3-manifold into PSL(N, C). It was used, for example, to generate the tables of representations at ptolemy.unhyperbolic.org.

The ptolemy module can use magma for the computations necessary to find the representations or it can automatically retrieve the necessary computations from a database we provide that contains the computations for all manifolds and N at ptolemy.unhyperbolic.org. In particular, the database includes the computations for all orientable census manifolds for PSL(2, C) and the examples below work without magma for these manifolds. This makes it useful for people who do not have magma (see here).

The ptolemy module is still under development. Please report bugs or email suggestions to: enischte at gmail dot com.

Examples of what the Ptolemy module can do:

The ptolemy module tries to have powerful and flexible functionality but at the same time be easy to use. For example, we can get the volumes of boundary-unipotent PSL(2, C)-representations of m011 in just one line:

>>> Manifold("m011").ptolemy_variety(2,'all').retrieve_solutions().volume_numerical()
[[[-4.30211422042248 E-16, -0.942707362776931, 0.942707362776931]],
 [[4.64255370258293 E-15, 3.94215909915729 E-15, -2.78183391239608, 2.78183391239608]]]

Here, we see that m011 has a representation that is not Galois-conjugate to the geometric representation and that has the volume 0.9427... of the Weeks manifold.

Remark: Please check your Internet connection if the above example does not work.

Documentation