# Orbits¶

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.

## Installation¶

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.