We perform full-magnetohydrodynamics simulations on various initially helical configurations and show that they reconfigure into a state where the magnetic field lines span nested toroidal surfaces. This relaxed configuration is not a Taylor state, as is often assumed for relaxing plasma, but a state where the Lorentz force is balanced by the hydrostatic pressure, which is lowest on the central ring of the nested tori. Furthermore, the structure is characterized by a spatially slowly varying rotational transform, which leads to the formation of a few magnetic islands at rational surfaces. We then obtain analytic expressions that approximate the global structure of the quasistable linked and knotted plasma configurations that emerge, using maps from S 3 to S 2 of which the Hopf fibration is a special case. The knotted plasma configurations have a highly localized magnetic energy density and retain their structure on time scales much longer than the Alfvénic time scale.

Since its discovery in 1931, the Hopf fibration has played an important role in physics in seemingly unrelated situations ranging from qubits to Taub-NUT spaces in general relativity [1]. Recently the structure of this fibration has been considered in relation to solutions of the source-free Maxwell equations and led to linked and knotted forms of electromagnetic fields [2, 3]. In twistor theory, it was the structure of the Hopf fibration that gave a twistor its name [4]. Here we present a deeper correspondence between a (non-null) twistor and the knotted electromagnetic fields, in which the Poynting vector plays a central role.

We derive a superpotential for null electromagnetic fields in which the field line structure is in the form of an arbitrary torus knot. These fields are shown to correspond to single copies of a class of anti-self-dual Kerr-Schild spacetimes containing the Sparling-Tod metric. This metric is the pure Weyl double copy of the electromagnetic Hopfion, and we show that the Eguchi-Hanson metric is a mixed Weyl double copy of this Hopfion and its conformally inverted state. We formulate two conditions for electromagnetic fields, generalizing torus knotted fields and linked optical vortices, that, via the zero rest mass equation for spin 1 and spin 2, defines solutions of linearized Einstein's equation possessing a Hopf fibration as the curves along which no stretching, compression or precession will occur. We report on numerical findings relating the stability of the linked and knotted zeros of the Weyl tensor and their relation to linked optical vortices. † s.sabharwal@umail.leidenuniv.nl ‡ Dalhuisen@physics.leidenuniv.nl 1 arXiv:1904.06030v3 [hep-th]