Optical knots and contact geometry II.From Ranada dyons to transverse and cosmetic knots

TL;DR

This paper extends the classification of knotted and linked structures in gauge fields, demonstrating that only iterated torus knots and links can be dynamically generated in boundaryless ideal fluids, with implications for particle-knot/link correspondence in physics.

Contribution

It provides a comprehensive classification of knotted gauge field structures, linking fluid dynamics and electrodynamics through Abelian reduction and recent progress on the Moffatt conjecture.

Findings

Only iterated torus knots and links can be dynamically generated in boundaryless ideal fluids.

The study completes the classification of knotted structures in gauge fields.

Results support the particle-knot/link correspondence in high energy physics.

Abstract

Some time ago Ranada (1989) obtained new nontrivial solutions of the Maxwellian gauge fields without sources. These were reinterpreted in Kholodenko (2015a) (part I) as particle-like (monopoles, dyons, etc.). They were obtained by the method of Abelian reduction of the non-Abelian Yang-Mills functional. The developed method uses instanton-type calculations normally employed for the non-Abelian gauge fields. By invoking the electric-magnetic duality it then becomes possible to replace all known charges/masses by the particle -like solutions of the source-free Abelian gauge fields. To employ these results in high energy physics, it is essential to to extend Ranada's results by carefully analysing and classifying all dynamically generated knoted/linked structures in gauge fields, including those discovered by Ranada. This task is completed in this work. The study is facilitated by the recent progress made in solving the Moffatt conjecture. Its essence is stated as follows: in steady incompressible Euler-type fluids the streamlines could have knots/links of all types. By employing the correspondence between the ideal hydrodynamics and electrodynamics discussed in part I and by superimposing it with the already mentioned method of Abelian reduction, it is demonstrated that in the absence of boundaries only the iterated torus knots and links could be dynamically generated. Obtained results allow to develop further particle-knot/link correspondence studied in Kholodenko (2015b)