veering 0.2

Taut and veering triangulations

Veering

Python code for working with transverse taut and veering ideal triangulations; implemented by Anna Parlak, Saul Schleimer, and Henry Segerman. The taut and veering polynomials are defined by Michael Landry, Yair Minsky and Sam Taylor. We thank Nathan Dunfield for many helpful comments (and for some code).

Installation

To install Veering inside Sage via the command line type:

sage -pip install git+https://github.com/henryseg/Veering

If you wish to install in Python, replace sage -pip by pip in the above command.

Essentially all of the veering code relies on regina; some of it relies on snappy and some 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

Testing

For a sanity check do, run in a sage console:

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

Usage

As an example, start a sage sessiond and type:

sage: from veering.file_io import parse_data_file
sage: veering_isosigs = parse_data_file('veering_census.txt')

The list veering_isosigs now contains all taut isomorphism signatures for the veering triangulations with at most 16 tetrahedra. These are ordered lexicographically.

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

This is the taut isomorphism signature for the only known 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 the positions of the edges with dihedral angle pi in each tetrahedron.

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

This taut structure is layered; we deduce that the figure eight knot is a fibered knot.

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

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 = veering_isosigs[257]
sage: from veering import taut_polytope
sage: taut_polytope.cone_in_homology(sig)
[N(1, -1), N(1, 1)]

The cone of homology classes carried by the veering triangulation veering_isosigs[257] 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.

Webpage

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

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

Citation

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

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

Contact

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

Licence

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