Torus knots as Hopfions

TL;DR

This paper establishes a link between torus knots and Hopfions by finding stable solutions in the Faddeev-Skyrme model, showing Hopfions can be classified by torus knot types based on their topological properties.

Contribution

It introduces a novel classification of Hopfions using torus knot types through explicit stable solutions in the Faddeev-Skyrme model.

Findings

Torus knots correspond to specific Hopfion solutions with charge PQ.

Hopfions can be classified by torus knot type based on their topological structure.

Stable, static solutions of the Faddeev-Skyrme model are constructed with a ferromagnetic potential.

Abstract

We present a direct connection between torus knots and Hopfions by finding stable and static solutions of the extended Faddeev-Skyrme model with a ferromagnetic potential term. (P,Q)--torus knots consisting of |Q| sine-Gordon kink strings twisted P/Q times into the poloidal cycle along the toroidal cycle on a toroidal domain wall carry the Hopf charge PQ, which demonstrates that Hopfions can be further classified according to torus knot type.