Untangling knots via reaction-diffusion dynamics of vortex strings
Abstract
We introduce and illustrate a new approach to the unknotting problem via the dynamics of vortex strings in a nonlinear partial differential equation of reaction-diffusion type. To untangle a given knot, a Biot-Savart construction is used to initialize the knot as a vortex string in the FitzHugh-Nagumo equation. Remarkably, we find that the subsequent evolution preserves the topology of the knot and can untangle an unknot into a circle. Illustrative test case examples are presented, including the untangling of a hard unknot known as the culprit. Our approach to the unknotting problem has two novel features, in that it applies field theory rather than particle mechanics and uses reaction-diffusion dynamics in place of energy minimization.
Cite
@article{arxiv.1604.04542,
title = {Untangling knots via reaction-diffusion dynamics of vortex strings},
author = {Fabian Maucher and Paul Sutcliffe},
journal= {arXiv preprint arXiv:1604.04542},
year = {2016}
}
Comments
5 pages, 4 figures, 3 movies. To appear in Phys. Rev. Lett