Orbits is free software licensed under the GNU Public license, version 2 or higher, and it is provided without warranty of any kind.

In a remarkable paper by Ian Agol, Joel Hass and Bill Thurston it was shown that the problem of deciding whether a knot has a genus \(g\) spanning surface is NP complete. Along the way, the authors described an algorithm for solving the following problem:

  • Given an equivalence relation on the set \(J = \{1, ..., N\}\) which
    is generated by \(k\) isometric pairings between subintervals of \(J\),
    compute the number of equivalence classes.

They showed that the running time of the algorithm is amazingly fast – bounded by a polynomial in \(k \log N\).

Orbits is a python module which implements their algorithm.


Orbits is a simple python module consisting of a single file. To install it run

python setup.py install

or, on linux,

sudo python setup.py install

A simple example computation is provided in the file example.py.

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