title: 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.
authors.affiliation: Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom.|Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom.
authors.forename: Fabian
authors.initials: F
authors.lastname: Maucher
authors.affiliation: Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom.
authors.forename: Paul
authors.initials: P
authors.lastname: Sutcliffe