LETTERS PUBLISHED ONLINE: 7 MARCH 2016 | DOI: 10.1038/NPHYS3679 How superfluid vortex knots untie Dustin Kleckner1*†, Louis H. Kauman2 and William T. M. Irvine1* Knots and links often occur in physical systems, including Here, we report on a systematic study of the behaviour of all prime shaken strands of rope1 and DNA (ref.2), as well as the topologies up to nine crossings by simulating isolated quantum more subtle structure of vortices in fluids3 and magnetic vortex knots in the Gross–Pitaevskii equation (GPE, equation (1)). fields in plasmas4. Theories of fluid flows without dissipation The quantum counterpart of smoke rings in air, vortices in predict these tangled structures persist5, constraining the superfluids or superconductors arep line-like phase defects in the evolution of the flow much like a knot tied in a shoelace. quantum order parameter, .x/ D ρ.x/eiφ.x/, where ρ and φ are This constraint gives rise to a conserved quantity known as the spatially varying density and phase (Fig. 1e). The GPE is a useful helicity6,7, oering both fundamental insights and enticing model system for studying topological vortex dynamics: vortex lines possibilities for controlling complex flows. However, even small are easily identified, reconnections occur without divergences in amounts of dissipation allow knots to untie by means of ‘cut- physical quantities, and the behaviour of simple knots was recently and-splice’ operations known as reconnections3,4,8–11. Despite shown to be comparable to viscous fluid experiments12. the potentially fundamental role of these reconnections in In a non-dimensional form, the Gross–Pitaevskii equation is understanding helicity—and the stability of knotted fields more given by19: generally—their eect is known only for a handful of simple d i 2 2 12 D− r − j j −1 (1) knots . Here we study the evolution of 322 elemental knots dt 2 and links in the Gross–Pitaevskii model for a superfluid, and find that they universally untie. We observe that the centreline where in these units the quantized circulation around a single vortex helicity is partially preserved even as the knots untie, a remnant line is given by: Γ D H d` · u D 2π. The GPE has a characteristic of the perfect helicity conservation predicted for idealized length scale, known as the `healing length', ξ, which corresponds fluids. Moreover, we find that the topological pathways of to the size of the density-depleted region around each vortex core untying knots have simple descriptions in terms of minimal (ξ D 1 in our non-dimensional units if the background density is ρ two-dimensional knot diagrams, and tend to concentrate in 0 D1). states which are twisted in only one direction. These results Producing a knotted vortex in a superfluid model requires the have direct analogies to previous studies of simple knots in computation of a space-filling complex function whose phase field several systems, including DNA recombination2 and classical contains a knotted defect. This challenging step has restricted fluids3,12. This similarity in the geometric and topological previous studies to one family of knots in a specific geometry8. evolution suggests there are universal aspects in the behaviour By numerically integrating the flow field of a classical fluid vortex, of knots in dissipative fields. we produce phase fields with defects (vortices) of any topology or Tying a knot has long been a metaphor for creating stability, and geometry12 (Fig. 1e and Supplementary Movie 1), enabling us to for good reason: untangling even a common knotted string requires study the evolution of every prime knot and link with nine or fewer either scissors or a complicated series of moves. This persistence has crossings, n≤9. important consequences for filamentous physical structures such as To construct initial shapes for the different topologies, we begin DNA, the behaviour of which is altered by knots and links9,13. An with the `ideal' form for each knot, equivalent to the shape of the analogous effect can be seen in physical fields, for example, magnetic shortest knot tied in a rope of thickness r0 (Fig. 1b–d; ref. 20). These fields in plasmas or vortices in fluid flow; in both cases knots never canonical shapes are known to capture key aspects of the knot type untie in idealized models, giving rise to new conserved quantities6,14. as well as approximating the average properties of random knots21. At the same time, there are numerous examples in which forcing For each ideal shape we consider three different overall scalings /ξ real (non-ideal) physical systems causes them to become knotted: with respect to the healing length: r0 D f15, 25, 50g. To break any vortices in classical or superfluid turbulence15,16, magnetic fields in symmetries of the shape and to check for robustness of our results the solar corona4, and defects in condensed matter phases10. This we also consider four randomly distorted versions of each knot with ξ presents a conundrum: why doesn't everything get stuck in a tangled n≤8 at a scale of r0 D15 (see Methods for a detailed description of web, much like headphone cords in a pocket1? the construction). In all of these systems, `reconnection events' allow fields to Figure 2a and Supplementary Movie 2 show the evolution untangle by cutting and splicing together nearby lines/structures of a 6-crossing knot, K6-2, as it unties. (We label links and (Fig. 1a; refs3,4,8–11). As a result, the balance of knottedness, and knots using a generalized notation following the `Knot Atlas', its fundamental role as a constraint on the evolution of physical http://katlas.org.) The knot can be seen to deform towards a series of systems, depends critically on understanding if and how these vortex reconnections that progressively simplify the knot until only mechanisms cause knots to untie. unknotted rings (unknots) remain. This behaviour has previously Previous studies of the evolution of knotted fields have been been observed for a handful of simple knots and links; here we restricted to relatively simple topologies or idealized dynamics3,9,17,18. find the same behaviour in all of the 1,458 simulated vortex knots. 1James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA. 2Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607, USA. †Present address: University of California, Merced, Merced, California 95343, USA. *e-mail: [email protected]; [email protected] 650 NATURE PHYSICS | VOL 12 | JULY 2016 | www.nature.com/naturephysics © 2016 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS3679 LETTERS a RReconnection b Ideal knot c r0 = drope de n = 2 [1] 3 [1] 4 [3] 5 [3] L2a1 K3-1 K5-2 L4a1 6 [13] 7 [21] 8 [81] 9 [199] Phase (φ ) π π π π L6n1 K7-1 L8a21 K9-17 0 1/2 3/2 2 f K3-1 L4a1 K4-1 K9-17 n = 3, w = +3 n = 4, w = −4 n = 4, w = 0 n = 9, w = +1 Figure 1 | Reconnection events and vortex knots. a, A schematic of a vortex reconnection event, in this case converting a trefoil knot (K3-1) to a pair of linked rings (L2a1). b, An ‘ideal’, or minimum rope-length, trefoil knot. c, Using the centreline of an ideal knot provides a consistent, uniform geometry for any knot or link; nearby strands are exactly spaced by the rope diameter, drope, which becomes the characteristic radius, r0, of the loops which compose the knot. d, Example ideal configurations of topologies with dierent minimal crossing number, n. The number of topologies excluding mirrored pairs is indicated in square brackets. e, A 2D slice of the phase field of a superfluid order parameter with a knotted vortex line (light blue). f, Example minimal knot diagrams; in each case the topology cannot be represented by a simpler planar diagram. The chirality of each crossing is indicated. Wefurther note that during the evolution of any sufficiently complex would include a term proportional to the twist inside the core (see knot, strongly distorted forms of simpler vortex knots are produced, Methods for a discussion of twist in the context of superfluid cores). which all in turn exhibit similar untying dynamics to their more Three general trends can be clearly discerned from our results: ideally shaped counterparts. the timescale for unknotting is determined predominantly by the /ξ We quantify the vortex dynamics by computing the overall scale of the knot, r0 , where r0 is the rope thickness of dimensionless length, vortex energy and helicity, as a function of the ideal shape used to generate the initial state (Fig. 3a–d); the time (Fig. 2c–e and Supplementary Fig. 2, see Methods for details). helicity is not simply dissipated, but rather converted from links The vortex energy, computed from the shape of the superfluid phase and knots into helical coils, with an efficiency that depends on scale defect, measures the energy associated with the vortical flow, as (Fig. 3e–h); and the vortex lines stretch by ∼20% as they untie, opposed to sound waves. The total combined energy (from vortices even though the vortex energy decreases slightly. We note that the and sound waves) is conserved in the GPE unless a dissipative term vortex energy changes through reconnections, as some energy is is added; we do not include one here. converted to sound waves (in line with previous observations of The non-dimensional `centreline helicity', h—which measures the colliding rings23). Interestingly, all of these results are apparently, 6,7,12,22 total linking, knotting and coiling in the field—is given by : on average, independent of knot complexity: for the same scale, r0, simple knots untie just as quickly as complicated ones, and lose the X X same relative amount of helicity and vortex energy (Supplementary hD Lkij C Wri (2) i6Dj i Fig.