Veerer is a package for SageMath to deal with veering triangulations of surfaces and their associated flat structures. It can in particular be used to provide representatives of pseudo-Anosov mapping classes of surfaces. The theoretical background is based on ideas of I. Agol and F. Guéritaud, and is developed in:

M. Bell, V. Delecroix, V. Gadre, R. Gutiérrez-Romo, S. Schleimer, "Coding Teichmüller flow using veering triangulations", arXiv:1909.00890.

As in Flipper, edges of a triangulation are labeled with the integers 0, 1, ..., n-1. Each edge comes with an orientation and the edge opposite to i is labelled ~i (so -1, -2, etc. are respectively opposite edges to 0, 1, etc.). To input a triangulation, you must provide a list of triangles, each triangle being 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 a 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 use Veerer inside SageMath, the flat structure can be displayed with:

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

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

$ sage -pip install -e .

and then run:

$ sage -t --force-lib veerer/

Or:

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

Go to the docs directory 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 convert veerer_demo.rst into veerer_demo.ipynb you need to have available on your computer

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

Then do:

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

If you installed rst2ipynb using the --user option of pip, the executables might be 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