Knots, Braids and Hedgehogs from the Eikonal Equation

TL;DR

This paper explores solutions to the three-dimensional complex eikonal equation, revealing new topological configurations such as braided strings and hedgehogs, which extend the understanding of eikonal knots and their relation to physical models.

Contribution

It introduces new topological solutions to the complex eikonal equation, including braided strings and hedgehogs, expanding the class of known configurations beyond torus knots.

Findings

Discovery of braided open strings and hedgehogs as solutions.

Identification of cylindric string solutions on arbitrary cylindrical surfaces.

Relevance to approximating Faddeev-Niemi hopfions in physics.

Abstract

The complex eikonal equation in the three space dimensions is considered. We show that apart from the recently found torus knots this equation can also generate other topological configurations with a non-trivial value of the $\pi_2(S^2)$ index: braided open strings as well as hedgehogs. In particular, cylindric strings i.e. string solutions located on a cylinder with a constant radius are found. Moreover, solutions describing strings lying on an arbitrary surface topologically equivalent to cylinder are presented. We discus them in the context of the eikonal knots. The physical importance of the results originates in the fact that the eikonal knots have been recently used to approximate the Faddeev-Niemi hopfions.