The ptolemy module ================== .. automodule:: snappy.ptolemy 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 :ref:`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: * Give the :ref:`equations for the reduced Ptolemy varieties` to find all :ref:`generically decorated ` and :ref:`boundary-unipotent ` representations into SL(*N*, **C**). * Same for representations into PSL(*N*, **C**) using :ref:`Obstruction classes `. * Retrieve the :ref:`exact ` and :ref:`numerical ` solutions for these varieties. * Compute the matrices for these representations :ref:`exactly ` or :ref:`numerically ` given a generator of or word in the fundamental group. * Compute the :ref:`traces ` for these matrices. * Compute the :ref:`trace field ` of the representation. * Compute the :ref:`boundary holonomoy `. * Compute the corresponding :ref:`shape parameters/cross ratios ` for a representation. * Compute the :ref:`volume ` and :ref:`complex volume ` (volume + i Chern-Simons) for a representation. * Compute solutions to the Ptolemy varieties :ref:`using magma or sage `. * Find :ref:`positively dimensional components ` of the Ptolemy variety. * Find :ref:`positively dimensional families ` of boundary-unipotent representations. 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 ------------- .. toctree:: :maxdepth: 2 ptolemy_prelim ptolemy_examples1 ptolemy_examples2 ptolemy_examples3 ptolemy_examples4 ptolemy_classes