Untangling knots via reaction-diffusion dynamics of vortex strings

TL;DR

This paper presents a novel reaction-diffusion based method using vortex string dynamics in a PDE to solve the unknotting problem, successfully untangling knots while preserving topology, and offering a new field theory approach.

Contribution

It introduces a new reaction-diffusion PDE approach for unknotting, replacing traditional energy minimization with vortex string dynamics in field theory.

Findings

Successfully untangles knots including complex unknots.

Preserves knot topology during evolution.

Demonstrates effectiveness with illustrative examples.

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.