Maxwell's Theory of Solid Angle and the Construction of Knotted Fields

TL;DR

This paper systematically describes the solid angle function for constructing knotted fields from curves in three-dimensional space, linking geometry, topology, and physical applications.

Contribution

It introduces a geometric framework for knotted field construction using the solid angle, connecting it to curve properties like writhe and providing computational tools.

Findings

Solid angle induces a natural framing related to writhe.

Provides explicit formulas and C code for constructing knotted fields.

Demonstrates applications in excitable media and liquid crystals.

Abstract

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in $\mathbb{R}^3$. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighborhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.