Harmonic field in knotted space

TL;DR

This paper investigates harmonic fields within knotted tubes, revealing how topology influences field patterns and transitions, with implications for controlling physical fields through surface topology manipulation.

Contribution

It demonstrates the uniqueness of harmonic fields in knotted domains and connects topology with physical field organization, extending insights to liquid crystal textures.

Findings

Harmonic fields are unique in knotted tubes.

Topology drives bipolar to vortex pattern transitions.

Results applicable to controlling physical fields via surface topology.

Abstract

Knotted fields enrich a variety of physical phenomena, ranging from fluid flows, electromagnetic fields, to textures of ordered media. Maxwell's electrostatic equations, whose vacuum solution is mathematically known as a harmonic field, provide an ideal setting to explore the role of domain topology in determining physical fields in confined space. In this work, we show the uniqueness of a harmonic field in knotted tubes, and reduce the construction of a harmonic field to a Neumann boundary value problem. By analyzing the harmonic field in typical knotted tubes, we identify the torsion driven transition from bipolar to vortex patterns. We also analogously extend our discussion to the organization of liquid crystal textures in knotted tubes. These results further our understanding about the general role of topology in shaping a physical field in confined space, and may find applications in the control of physical fields by manipulation of surface topology.