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.

Stable and unstable vortex knots in excitable media

“…We strengthen these results, and for the unknot those of Ref. [13], by testing the bulk untangling dynamics of all of these knots with a wide variety of initial conditions -in the case of the unknot, a far greater variety than has been used previously. We find that in the bulk a generic unknot, trefoil or figure eight simplifies to a canonical form, but that the wave slapping mechanism at play for large N can cause rates of convergence to vary dramatically.…”

“…So far, the striking knot simplification has been reported only for the unknot [13] -reducing three examples of tangled, but unknotted, curves to a geometric circle of fixed average radius -and, in two examples, the trefoil [13,14]. At higher crossing number the behaviour appears to be more complicated.…”

Length of excitable knots

“…To date, the only evidence [8] over long time scales for this conjecture has been restricted to the simplest nontrivial knot, the trefoil knot, and some very recent results on the untangling of unknots [9]. Here we provide evidence for this conjecture by presenting solutions for all torus knots up to crossing number 11.…”

“…These results reveal that the amplitude of the oscillation in the length decreases as p increases but increases as q increases. It has been shown that FitzHugh-Nagumo flow is able to untangle unknots with a complex initial conformation [9], but the physical mechanism that underpins this process remains elusive. Although the untangling dynamics is highly nontrivial, a phenomenological understanding, applicable in the regime of slight curvature and twist, can be obtained by combining positive filament tension [16,17] with a short-range repulsion between vortex strings.…”

“…The center of the vortex is the point at which |∇u × ∇v| is maximal, and this quantity is localized in the vortex core [13]. To provide initial conditions for a vortex string of an arbitrary knot, given by any nonintersecting closed curve K, we apply the method introduced in [9] and adapted from [14]. This involves computing the initial fields u(r,0) and v(r,0) from a scalar potential for a vector field defined by a Biot-Savart integral along the curve K. For most of our investigation we will restrict the discussion to the case where K is a torus knot.…”