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 analytical 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 (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 known knot types. Furthermore,it allows to reinterpret both the electric and magnetic charges purely topologically thus opening the possibility of treatment of masses and charges in Yang-Mills and gravity theories also topologically. According to(now proven) Thurston's geometrization conjecture complements of all knots/links in S^3 are spaces of positive, zero or negative curvature. This means that spaces around our topological masses/charges are also curved. This fact is essential for design of purely topological theories of gravity, electromagnetism and strong/weak interactions