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Untangling Knots Via Reaction-Diffusion Dynamics of Vortex Strings

Fabian Maucher and Paul Sutcliffe
Phys. Rev. Lett. 116, 178101 – Published 27 April 2016
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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.

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  • Received 9 March 2016

DOI:https://doi.org/10.1103/PhysRevLett.116.178101

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

General Physics

Authors & Affiliations

Fabian Maucher1,2,* and Paul Sutcliffe2,†

  • 1Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
  • 2Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom

  • *fabian.maucher@durham.ac.uk
  • p.m.sutcliffe@durham.ac.uk

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Issue

Vol. 116, Iss. 17 — 29 April 2016

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It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 3.0 License. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

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