New knotted solutions of Maxwell's equations

In this paper 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 obtain a wide class of solutions from the basic configuration, such as constant electromagnetic fields and plane-waves. We have introduced a covariant formulation of 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 the charge takes integer values.