We extend the definition of hopfions to include a class of spin-h fields and use this to introduce the electromagnetic and gravitational hopfions of different algebraic types. The fields are constructed through the Penrose contour integral transform, thus the singularities of the generating functions are directly related to the geometry of the resulting physical fields. We discuss this relationship and how the topological structure of the fields is related to the Robinson congruence. Since the topology appears in the lines of force for both electromagnetism and gravity, the gravito-electromagnetic formalism is used to analyze the gravitational hopfions and describe the time evolution of their tendex and vortex lines. The correspondence between fields of different spin results in analogous configurations based on the same topological structure. The null and type N fields propagate at the speed of light, while the non-null and type D fields radiate energy outward from the center. Finally we discuss the type III gravitational hopfion, which has no direct electromagnetic analog, but find that it still exhibits some of the characteristic features common to the other hopfion fields.

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.

We present a class of topological plasma configurations characterized by their toroidal and poloidal winding numbers, n t and n p , respectively. The special case of n t = 1 and n p = 1 corresponds to the Kamchatnov-Hopf soliton, a magnetic field configuration everywhere tangent to the fibers of a Hopf fibration so that the field lines are circular, linked exactly once, and form the surfaces of nested tori. We show that for n t ∈ Z + and n p = 1, these configurations represent stable, localized solutions to the magnetohydrodynamic equations for an ideal incompressible fluid with infinite conductivity. Furthermore, we extend our stability analysis by considering a plasma with finite conductivity, and we estimate the soliton lifetime in such a medium as a function of the toroidal winding number.

We show that the torus knot topology is inherent in electromagnetic and gravitational radiation by constructing spin-N fields based on this topology from the elementary states of twistor theory. The twistor functions corresponding to the elementary states admit a parameterization in terms of the poloidal and toroidal winding numbers of the torus knots, allowing one to choose the degree of linking or knotting of the associated field configuration. Using the gravitoelectromagnetic formalism, we show that the torus knot structure is exhibited in the tendex and vortex lines for the analogous linearized gravitational solutions. We describe the topology of the gravitational fields and its physical interpretation in terms of the tidal and frame drag forces of the gravitational field.