Python code (using regina, snappy, and sage) for working with transverse taut and veering ideal triangulations -- implemented by Anna Parlak, Saul Schleimer, and Henry Segerman. We thank Nathan Dunfield for many helpful comments (and for some code).

To install (or uninstall) veering inside Sage via the command line, using pip, type:

sage -pip install veering

or

sage -pip uninstall veering

For installation into your system's Python, replace sage -pip by pip3. Note that the github repository of veering contains further data and scripts which are not installed by pip.

Almost all of the veering code relies on regina; some of it relies on snappy and some relies on SageMath. Other parts rely on the Python vector graphics package pyx. Installation instructions for SageMath, snappy, and regina can be found at the following webpages:

https://doc.sagemath.org/html/en/installation/
https://snappy.math.uic.edu/installing.html
https://github.com/3-manifolds/regina_wheels

After installation start a sage session and run the following:

sage: import veering
sage: from veering import test_suite
sage: test_suite.run_tests()

Each test should take at most a few seconds.

As a simple example:

sage: census = veering.veering_census(); len(census)
87047

The veering census contains the 87047 taut isomorphism signatures of the veering triangulations with at most 16 tetrahedra. These are ordered lexicographically.

sage: sig = census[1]; sig
'cPcbbbiht_12'

This is the taut isomorphism signature for the only veering structure on the figure eight knot complement. The string before the underscore is the isomorphism signature for the triangulation; the string after the underscore records, for each tetrahedron, which two edges have dihedral angle pi; the other four edges have dihedral angle zero.

sage: from veering import taut_polytope
sage: taut_polytope.is_layered(sig)
True

This taut structure is layered; thus the figure-eight knot is fibered.

sage: from veering import taut_polynomial
sage: taut_polynomial.taut_polynomial_via_tree(sig)
a^2 - 3*a + 1
sage: taut_polynomial.taut_polynomial_via_tree(sig, mode = 'alexander')
a^2 - 3*a + 1
sage: from veering import veering_polynomial
sage: veering_polynomial.veering_polynomial(sig)
a^3 - 4*a^2 + 4*a - 1

The taut and veering polynomials are defined by Michael Landry, Yair Minsky and Sam Taylor. Note that the taut polynomial divides the veering polynomial; this is true in general. The taut polynomial of this veering triangulation is equal to the Alexander polynomial of the underlying manifold; this is not true in general.

sage: sig = census[257]; sig
'iLLLQPcbeegefhhhhhhahahha_01110221'
sage: taut_polytope.cone_in_homology(sig)
[N(1, -1), N(1, 1)]

The cone of homology classes carried by the veering triangulation iLLLQPcbeegefhhhhhhahahha_01110221 is spanned by the rays passing through (1,-1) and (1,1). Landry, Minsky, and Taylor proved that, if nonempty, this cone is equal to a cone on a (not necessarily top-dimensional) face of the Thurston norm ball. The chosen basis on H^1 is dual to the basis of H_1 used to compute the taut and veering polynomials.

For references, for information about the census, and for many diagrams, please see:

https://math.okstate.edu/people/segerman/veering.html

When citing the codebase, please use the following (updating the version number and the year).

@Misc{Veering,
    author = {Anna Parlak and Saul Schleimer and Henry Segerman},
    title = {veering x.y, code for studying taut and veering ideal triangulations},
    howpublished = {\url{https://github.com/henryseg/Veering}},
    year = {20zz},
}

Please contact us with any and all suggestions, questions, and/or corrections.

This work is in the public domain. See the LICENCE for details.