How superfluid vortex knots untie

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), offering both fundamental insights and enticing possibilities for controlling complex flows. However, even small amounts of dissipation allow knots to untie by means of 'cut-and-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 effect 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.