Knots in electromagnetism

Abstract

Maxwell equations in vacuum allow for solutions with a non-trivial topology in the electric and magnetic field line configurations at any given moment in time. One example is a space filling congruence of electric and magnetic field lines forming circles lying on the surfaces of nested tori. In this example the electric, magnetic and Poynting vector fields are orthogonal everywhere. As time evolves the electric and magnetic fields expand and deform without changing the topology and energy, while the Poynting vector structure remains unchanged while propagating with the speed of light. The topology is characterized by the concept of helicity of the field configuration. Helicity is an important fundamental concept and for massless fields it is a conserved quantity under conformal transformations. We will review several methods by which linked and knotted electromagnetic (spin-1) fields can be derived. A first method, introduced by A. Rañada, uses the formulation of the Maxwell equations in terms of differential forms combined with the Hopf map from the three-sphere S 3 to the two-sphere S 2. A second method is based on spinor and twistor theory developed by R. Penrose in which elementary twistor functions correspond to the family of electromagnetic torus knots. A third method uses the Bateman construction of generating null solutions from complex Euler potentials. And a fourth method uses special conformal transformations, in particular conformal inversion, to generate new linked and knotted field configurations from existing ones. This fourth method

On the Fibration Defined by the Field Lines of a Knotted Class of Electromagnetic Fields at a Particular Time

“…We have calculated the Euler potentials [13,17] of such fields, and used them to study the configuration of electric and magnetic field lines at t = 0. We have seen how the fibration defined by the Euler potentials at t = 0 consists of linked torus knots.…”

“…We will present and make use of the Euler potentials (see [17] and references therein). The field lines have the property of being tangent to the field vectors at any point of space.…”

“…This electromagnetic knot has been extensively studied [8,[12][13][14][15][16]. A few generalizations of the Hopf-Rañada solution have been found (see [17] and references therein for a review). For example, in [4] it was showed that electromagnetic fields proportional to the Hopf-Rañada solution satisfy Equations (1) and (2) only if the proportionality constant is an integer number, so the Hopf index is equal to n 2 .…”

“…This class contains in particular the cases discussed in the previous paragraph, i. e. the Hopf-Rañada electromagnetic knots. From the scalars maps φ 0 (r), θ 0 (r) we will obtain the Euler potentials [13,17] of the field at that instant. The Euler potentials define manifolds and the field lines belong to the tangent bundle of those surfaces.…”

On the existence of the field line solutions of the Einstein–Maxwell equations

“…The importance of the electromagnetic fields in terms of field lines is emphasised by a large range of phenomena in which they seem to be present: in fluid physics [23,24], atmospheric physics [25], liquid crystals [26], plasma physics [27], optical vortices [28,29] and superconductivity [30]. (For a review of the subject and some of its applications see [31] and the references therein).…”