# Braids

## Description

The code available for download here was written by
Andrew Geng for a project that attempted to find
differently-knotted surfaces S in D^{4}
bounded by the same transverse link in ∂D^{4}.

This work was done under the mentorship of Paul Seidel
and supported by a grant from the Massachusetts Institute of Technology.

## Capabilities

- An implementation of braid arithmetic,
following the description given in [2].
- Functions for working with presentations of groups
and counting homomorphisms.
- Functions for manipulating braid factorizations using Hurwitz moves
(see [1]).

## Requirements

- Python 2.6 or 2.7
- SymPy ≥ 0.6.0, only needed
for counting homomorphisms to GL(F
_{p}^{n}).

## Download

## References

- D. Auroux, V. Kulikov, V. Shevchishin,
"Regular Homotopy of Hurwitz Curves",
*Izv. Math.* 68 (2004), 521–542.
- J. Cha et al, "An Efficient Implementation of Braid Groups",
*Advances in Cryptology: Proceedings of ASIACRYPT 2001,
Lecture Notes in Computer Science* (2001), 144–156.