The Electrodynamics of Torus Knots

Abstract

One of the major drawbacks for application of knot theory to electromagnetics has been the lack of available parameterizations which can be used to mathematically describe knotted curves. This is because knots have traditionally been studies within a topological context where parameterizations for the curves are not generally required. However, in order to successfully characterize the electromagnetic radiation and scattering properties of knots using Maxwell's equations, it is advantageous to develop parameterizations which can be used to geometrically describe the curves of these knots. This paper introduces such parameterizations for a family of knots known as (p,q)-torus knots. These knots reside on the surface of a standard torus, thereby making it possible to readily obtain useful parameterizations to describe them. The well-known trefoil is one important example of a (p,q)-torus knot.