Knots in the Skyrme-Faddeev model

TL;DR

This paper numerically computes and analyzes topological soliton solutions in the Skyrme-Faddeev model, revealing new knotted configurations and their energy properties for various Hopf charges.

Contribution

It introduces a comprehensive numerical approach to find and classify knotted solitons in the Skyrme-Faddeev model, including many new solutions and insights into their energy minima.

Findings

Numerous new knotted solutions for Hopf charges up to 16.

Torus knots are often local minima, with linked solutions sometimes being global minima.

Computed energies align with Ward's energy bound conjecture.

Abstract

The Skyrme-Faddeev model is a modified sigma model in three-dimensional space, which has string-like topological solitons classified by the integer-valued Hopf charge. Numerical simulations are performed to compute soliton solutions for Hopf charges up to sixteen, with initial conditions provided by families of rational maps from the three-sphere into the complex projective line. A large number of new solutions are presented, including a variety of torus knots for a range of Hopf charges. Often these knots are only local energy minima, with the global minimum being a linked solution, but for some values of the Hopf charge they are good candidates for the global minimum energy solution. The computed energies are in agreement with Ward's conjectured energy bound.