Knotted Configurations with Arbitrary Hopf Index from the Eikonal Equation

TL;DR

This paper explores solutions to the complex eikonal equation in three dimensions, demonstrating it can produce multi-soliton configurations with any Hopf index, including knotted structures like trefoil knots, which may aid in modeling knotted solitons.

Contribution

It shows that the complex eikonal equation can generate multi-soliton configurations with arbitrary Hopf index, including non-toroidal and knotted structures, expanding the understanding of topological solitons.

Findings

Generated multi-soliton configurations with arbitrary Hopf index

Found knotted solutions such as trefoil knots

Suggested applications in modeling knotted solitons in the Faddeev-Niemi model

Abstract

The complex eikonal equation in $(3+1)$ dimensions is investigated. It is shown that this equation generates many multi soliton configurations with arbitrary value of the Hopf index. In general, these eikonal hopfions do not have the toroidal symmetry. For example, a hopfion with topology of the trefoil knot is found. Moreover, we argue that such solitons might be helpful in construction of approximated analytical knotted solutions of the Faddeev-Niemi model.