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Untangling knots via reaction-diffusion dynamics of vortex strings.

Maucher, F. and Sutcliffe, P.M. (2016) 'Untangling knots via reaction-diffusion dynamics of vortex strings.', Physical review letters., 116 (17). p. 178101.


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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Publisher statement: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.
Date accepted:14 April 2016
Date deposited:29 April 2016
Date of first online publication:27 April 2016
Date first made open access:29 April 2016

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