Knotted Topological Phase Singularities of Electromagnetic Field

TL;DR

This paper investigates knotted topological phase singularities in electromagnetic fields, revealing their Hopf invariant relates to linking and self-linking numbers, and discusses their conservation during topological processes.

Contribution

It introduces a topological current framework to analyze knotted RS vortices and links the Hopf invariant to their linking and self-linking numbers, advancing understanding of electromagnetic topological structures.

Findings

Hopf invariant equals sum of linking and self-linking numbers.

Conservation of Hopf invariant during vortex splitting, merging, and intersection.

Topological current theory applied to electromagnetic field singularities.

Abstract

In this paper, knotted objects (RS vortices) in the theory of topological phase singularity in electromagnetic field have been investigated in details. By using the $\phi$-mapping topological current theory proposed by Prof. Duan, we rewrite the topological current form of RS vortices and use this topological current we reveal that the Hopf invariant of RS vortices is just the sum of the linking and self-linking numbers of the knotted RS vortices. Furthermore, the conservation of the Hopf invariant in the splitting, the mergence and the intersection processes of knotted RS vortices is also discussed.