veerer is a Python module to deal with veering triangulations of surfaces and their associated flat structures. It can in particular be used to provides representatives of pseudo-Anosov mapping classes of surfaces. The theoretical background is developed in M. Bell, V. Delecroix, V. Gadre, R. Gutiérrez-Romo, S. Schleimer "Coding Teichmüller flow using veering triangulations" (https://arxiv.org/abs/1909.00890) and is based on ideas of I. Agol and F. Guéritaud.

To install the module you need Python (preferably version 3 but Python 2 is supported). Computations involving polytopes are only available if the Python module pplpy is available (see https://gitlab.com/videlec/pplpy). It is installed by default with SageMath from version 8.7. Additional features are available if this module is used inside SageMath (https://www.sagemath.org/).

As with https://github.com/MarkCBell/flipper, edges of a triangulation are labeled with the integers 0, 1, ..., n-1. Each edge come with an orientation and the edge opposite to i is labelled ~i (that is -1 is the opposite of 0, -2 of 1, etc). To input a triangulation, you must provide a list of triangles, each triangle beeing a triple of oriented edges, and a list of colours:

>>> from veerer import *
>>> T = VeeringTriangulation([(0,1,2), (-1,-2,-3)], [RED, RED, BLUE])
>>> T.is_core()
True

Since the above example is a core triangulation it admits flat realization. One can be computed by taking the middle of the barycenter of the vertices of the polytope parametrizing the flat structures:

>>> F = T.flat_structure_middle()

If you veerer inside SageMath, the flat structure could be displayed with:

>>> F.plot(vertical_train_track=True)
>>> F.plot(horizontal_train_track=True)

To run the SageMath doctests, install the module with pip, typically:

$ sage -pip install . --user --force-reinstall

and then run:

$ sage -t --force-lib veerer/

Or:

$ sage -t --force-lib veerer/my_file.py

Go to the docs repository and then do:

$ sage -sh
$ make html

The documentation should be available under docs/build/ as HTML pages.

Typically you might want to use veerer_demo.rst as a Jupyter notebook. In order to do the conversion you need to have available on your computer

• rst2latex python-docutils
• pdflatex
• pandoc
• the Python module rst2ipynb
• the Python module nbconvert

Then do:

$ export FILE_PREFIX="veerer_demo"
$ rst2ipynb --kernel=sagemath veerer_demo.rst veerer_demo.ipynb

If you did install rst2ipynb using the --user option of pip the executables are possibly installed in $HOME/.local/bin. In which case you should first make the system aware of this via:

$ PATH=$PATH:$HOME/.local/bin

• Mark Bell
• Vincent Delecroix
• Saul Schleimer