New knotted solutions of Maxwell's equations

Abstract

In this note we have further developed the study of topologically non-trivial solutions of vacuum electrodynamics. We have discovered a novel method of generating such solutions by applying conformal transformations with complex parameters on known solutions expressed in terms of Bateman's variables. This has enabled us to get a wide class of solutions from the basic configuration like constant electromagnetic fields and plane-waves. We have introduced a covariant formulation of the Bateman's construction and discussed the conserved charges associated with the conformal group as well as a set of four types of conserved helicities. We have also given a formulation in terms of quaternions. This led to a simple map between the electromagnetic knotted and linked solutions into flat connections of SU (2) gauge theory. We have computed the corresponding Chern-Simons charge in a class of solutions and it takes integer values.

Properties of null knotted solutions to Maxwell's equations

“…To create our fields we follow Bateman's construction 5 , as demonstrated clearly by Hoyos et al 7 and Irvine et al 3 We use rational units throughout with wave vectors set equal to unity in the free field Maxwell equations. To ensure the equations have a high degree of symmetry, the fields are redefined as follows,…”

“…The mathematical construction 9,10 of the Hopf-fibration is via the method of stereographic projection from the four dimensional spherical space, S 3 , into three dimensional Euclidean space, R 3 . The equations for the Hopf-fibration are 7 ,…”

“…Field line plots of the knotted and linked electromagnetic vector fields become more convoluted the more complicated the knotting becomes. For some vector fields, it is possible to have an orthogonal set of vector fields that are constructed of gradients of underlying scalar fields 6,7 . These underlying scalar fields become useful in the visualisation of our knotted solutions as they illuminate the cores of the tori twisting around each other, and clarify the different knottings (See figure 2).…”

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

“…More recent studies have focused on other features of the electromagnetic fields in terms of field lines: in [8,9] the authors studied the connection between the knotted and linked solutions, the dynamics of the electric charges in this background and of the knotted fields were investigated in [11]- [14] and the topological quantization was discussed in [15]- [18]. The generalization of the above solutions to the torus knot topology was given in [19]- [21] and field lines and Hopf solutions were constructed in the nonlinear electromagnetism in [22]- [24]. 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].…”

Knotted optical vortices in exact solutions to Maxwell's equations

“…The discovery of the electromagnetic Hopf field by Ranada [1][2][3][4] initiated further studies of knotted electromagnetic field lines [5][6][7][8][9][10][11]. This led to the formulation of a family of knotted electromagnetic fields encoding torus knots by Kedia et al [8].…”