Optical knots and contact geometry I. From Arnol’d inequality to Ranada’s dyons

Abstract

Recently there had been a great deal of activity associated with various schemes of designing both analytic and experimental methods describing knotted structures in electrodynamics and in hydrodynamics. The majority of works in electrodynamics were inspired by the influential paper by Ranada (Lett Math Phys 18:97–106, 1989) and its subsequent refinements. In this work and in its companion we analyze Ranada’s results using methods of contact geometry and topology. Not only our analysis allows us to reproduce his major results but in addition, it provides opportunities for considerably extending the catalog of the known/obtained knot types. In addition, it allows to reinterpret both the electric and magnetic charges purely geometrically thus opening the possibility of treatment of masses and charges in Yang–Mills and gravitational fields purely geometrically.