The electrodynamics of torus knots

The electromagnetic radiation and scattering properties of thin knotted wires are considered in this chapter. A special class of knots, called torus knots, are introduced for the purpose of this investigation. The parameterizations available for torus knots are used in conjunction with Maxwell's equations to formulate useful mathematical representations for the fields radiated by these knots. These representations are then used to derive simple closed-form far-field expressions for electrically small torus knots. The derivation of a new Electric Field Integral Equation (EFIE) suitable for analysis of toroidally knotted wires is also outlined in this chapter. It is also demonstrated that the well-known expressions for the electromagnetic fields radiated by a circular loop antenna (canonical unknot) as well as the linear dipole antenna may be obtained as degenerate forms of the more general torus knot field representations. Finally, a moment method technique is applied to model the backscattering from a threefold rotationally symmetric trefoil knot as a function of frequency, polarization, and incidence angle.