How superfluid vortex knots untie

Abstract

Knots and links often occur in physical systems, including shaken strands of rope 1 and DNA (ref. 2), as well as the more subtle structure of vortices in fluids 3 and magnetic fields in plasmas 4 . Theories of fluid flows without dissipation predict these tangled structures persist 5 , constraining the evolution of the flow much like a knot tied in a shoelace. This constraint gives rise to a conserved quantity known as helicity 6,7 , o ering both fundamental insights and enticing possibilities for controlling complex flows. However, even small amounts of dissipation allow knots to untie by means of 'cutand-splice' operations known as reconnections 3,4,8-11 . Despite the potentially fundamental role of these reconnections in understanding helicity-and the stability of knotted fields more generally-their e ect is known only for a handful of simple knots 12 . Here we study the evolution of 322 elemental knots and links in the Gross-Pitaevskii model for a superfluid, and find that they universally untie. We observe that the centreline helicity is partially preserved even as the knots untie, a remnant of the perfect helicity conservation predicted for idealized fluids. Moreover, we find that the topological pathways of untying knots have simple descriptions in terms of minimal two-dimensional knot diagrams, and tend to concentrate in states which are twisted in only one direction. These results have direct analogies to previous studies of simple knots in several systems, including DNA recombination 2 and classical fluids 3,12 . This similarity in the geometric and topological evolution suggests there are universal aspects in the behaviour of knots in dissipative fields.Tying a knot has long been a metaphor for creating stability, and for good reason: untangling even a common knotted string requires either scissors or a complicated series of moves. This persistence has important consequences for filamentous physical structures such as DNA, the behaviour of which is altered by knots and links 9,13 . An analogous effect can be seen in physical fields, for example, magnetic fields in plasmas or vortices in fluid flow; in both cases knots never untie in idealized models, giving rise to new conserved quantities 6,14 . At the same time, there are numerous examples in which forcing real (non-ideal) physical systems causes them to become knotted: vortices in classical or superfluid turbulence 15,16 , magnetic fields in the solar corona 4 , and defects in condensed matter phases 10 . This presents a conundrum: why doesn't everything get stuck in a tangled web, much like headphone cords in a pocket 1 ?In all of these systems, 'reconnection events' allow fields to untangle by cutting and splicing together nearby lines/structures ( Fig. 1a; refs 3,4,8-11). As a result, the balance of knottedness, and its fundamental role as a constraint on the evolution of physical systems, depends critically on understanding if and how these mechanisms cause knots to untie.Previous studies of the evolution of knotted fields have been restricted ...

Helicity within the vortex filament model

“…Therefore, whenever we refer to φ in what follows, we will assume this represents only this component of the phase. The direction of the constant phase (or velocity potential) of the complex wavefunction has also been proposed by others as a possible direction that can be used to define N for superfluid vortices161718. This choice for the vector N turns out to be equivalent to using the so-called Seifert framing.…”

“…In a microscopic model, such as that given by the Gross-Pitaevskii equation, they are made possible by the “quantum-pressure” term which acts to couple the incompressible and compressible modes of the system. Using the Gross-Pitaevskii model, the effect of reconnections on helicity has recently been analyzed in several papers1617181925.…”

Models of random knots

“…C. In current research, such knotting phenomena are theoretically analyzed, numerically simulated, and experimentally created or identified in various physical systems. To mention some examples: knotted vortices in classical fluid flow (Kleckner and Irvine 2013) and in superfluids (Hall et al 2016;Kleckner et al 2016), optical vortices in laser beams (Dennis et al 2010), magnetic fields in plasma (Berger 1999), superposition of states in quantum mechanics (Berry 2001), and also nonlinear waves in biological and chemical excitable media (Winfree and Strogatz 1984).…”

Length of excitable knots

“…Over 30 years ago it was conjectured [6] that nontrivial knots in a FitzHugh-Nagumo medium might preserve their topology and be remarkably immune to the reconnection events that untie knotted vortex strings in a generic way in most systems [7]. 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].…”