Field line winding of braided vector fields in tubular subdomains

TL;DR

This paper introduces a topological invariant called field line winding that uniquely classifies the entanglement of braided vector fields in tubular domains, with applications in physics and engineering.

Contribution

It provides a complete topological characterization of braided vector fields using field line winding, a new measure that generalizes field line helicity.

Findings

Field line winding uniquely classifies vector field topology.

The method applies to arbitrary tubular subdomains.

It distinguishes entanglement without conflating with field strength.

Abstract

Braided vector fields on spatial subdomains homeomorphic to the cylinder play a crucial role in applications such as solar and plasma physics, relativistic astrophysics, fluid and vortex dynamics, elasticity, and bio-elasticity. Often the vector field's topology -- the entanglement of its field lines -- is non-trivial, and can play a significant role in the vector field's evolution. We present a complete topological characterisation of such vector fields (up to isotopy) using a quantity called field line winding. This measures the entanglement of each field line with all other field lines of the vector field, and may be defined for an arbitrary tubular subdomain by prescribing a minimally distorted coordinate system. We propose how to define such coordinates, and prove that the resulting field line winding distribution uniquely classifies the topology of a braided vector field. The field line winding is similar to the field line helicity considered previously for magnetic (solenoidal) fields, but is a more fundamental measure of the field line topology because it does not conflate linking information with field strength.