Weaving knotted vector fields with tunable helicity

TL;DR

This paper introduces a method to construct divergence-free knotted vector fields from complex scalar functions, enabling the encoding of various knots and links with tunable helicity, useful for physical field modeling.

Contribution

It provides a systematic way to generate and analyze knotted vector fields with controllable helicity from complex scalar functions, including explicit computation methods.

Findings

Constructed knotted vector fields from complex scalar functions.

Enabled explicit calculation of helicity for these fields.

Generated examples with both non-zero and zero helicity.

Abstract

We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot and its generalizations. As finite-energy physical fields they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.