Faddeev-Hopf knots: Dynamics of linked un-knots

TL;DR

This paper numerically investigates the dynamics of Faddeev-Hopf knots, showing how linked un-knots relax into various energy minima depending on their charges and orientations, with visualizations of the configurations.

Contribution

It introduces a numerical study of Faddeev-Hopf knots' relaxation dynamics and visualizes the resulting energy-minimized configurations.

Findings

Linked un-knots relax into different minima based on charges and handedness.

The relaxation process depends on initial configurations and topological properties.

Visualizations of gauge-invariant iso-surfaces illustrate the knot configurations.

Abstract

We have studied numerically Faddeev-Hopf knots, which are defined as those unit-vector fields in $R^3$ that have a nontrivial Hopf charge and minimize Faddeev's Lagrangian. A given initial configuration was allowed to relax into a (local) minimum using the first order dissipative dynamics corresponding to the steepest descent method. A linked combination of two un-knots was seen to relax into different minimum energy configurations depending on their charges and their relative handedness and direction. In order to visualize the results we plot certain gauge-invariant iso-surfaces.