Untangling knots via reaction-diffusion dynamics of vortex strings

Maucher, F.; Sutcliffe, P.M.

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.

Citation

Maucher, F., & Sutcliffe, P. (2016). Untangling knots via reaction-diffusion dynamics of vortex strings. Physical Review Letters, 116(17), Article 178101. https://doi.org/10.1103/physrevlett.116.178101

• Journal Article Type: Article
• Acceptance Date: Apr 14, 2016
• Online Publication Date: Apr 27, 2016
• Publication Date: Apr 27, 2016
• Deposit Date: Apr 28, 2016
• Publicly Available Date: Apr 29, 2016
• Journal: Physical Review Letters
• Print ISSN: 0031-9007
• Electronic ISSN: 1079-7114
• Publisher: American Physical Society
• Peer Reviewed: Peer Reviewed
• Volume: 116
• Issue: 17
• Article Number: 178101
• DOI: https://doi.org/10.1103/physrevlett.116.178101
• Public URL: https://durham-repository.worktribe.com/output/1405945
• Related Public URLs: https://arxiv.org/abs/1604.04542