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 reanalyze 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 curved. This fact is essential for design of purely topological theories of gravity, electromagnetism and strong/weak interactions.

Knotted solutions for linear and nonlinear theories: Electromagnetism and fluid dynamics

“…More solutions were found in [17][18][19]. A map of magnetohydrodynamics to just fluid dynamics was used extensively (for instance, [20][21][22]), and one also was able to show that in magnetohydrodynamics a defined "velocity of lines of force" v p = ( E × H)/H 2 [23] can be measured, and in some cases ("frozen field condition") coincides with the velocity of the fluid transporting it (see for instance [22,24], the last also considering transporting the electromagnetic Hopfion).…”

Gravitoelectromagnetic Knot Fields

“…Equations (46) and (47) describe the dynamics of a test particle of given mass in the GEM knot background.…”

“…A general major motivation to study the topological abelian fields is to understand the relationship between their physical and mathematical properties. This is an active area of research in classical electromagnetism with significant progress made recently in this direction by the discovery and the analysis of a large set of new solutions of the field equations that generalize the electromagnetic Rañada fiels in the linear [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54] and non-linear electrodynamics [55][56][57][58]. From the mathematical point of view, several mathematical structures related to the topologically non-trivial fields have been explored in the recent literature among which are the twistors [42], rational functions [51,54], fibre bundles [59], space-time foliations [60] and generalized Finsler geometries [61].…”

Knotted solutions, from electromagnetism to fluid dynamics

“…We saw that B = ∇× A in electromagnetism is analogous to ω = ∇× v in a fluid, in terms of constructions that have nonzero Hopf index and give a nonzero helicity. An extension of this analogy was intensively used (see, for instance, [22][23][24]), from magneto-hydrodynamics, i.e. a fluid coupled to electromagnetism, to hydrodynamics only.…”

“…A connection that was much used (for instance, [22][23][24]) is between magnetohydrodynamics (motion of a fluid coupled to electromagnetism) and only fluid dynamics. One maps the magnetic field B to the vorticity ω, but the fluid velocity v remains v, which means that one restricts to the space of solutions where B = ω = ∇ × v. Then it was found that in magnetohydrodynamics one can define a "velocity of the lines of force" v p = ( E × H)/H 2 [25] and measure it, and under a "frozen field condition" v p can coincide with the velocity v of the fluid transporting it, e.g.…”