The Axial Electric Potential and Length of a Torus Knot

Physical knot theory, where knots are treated like physical objects, is important to many fields. One natural problem is to give a knot a uniform charge, and analyze the resulting electric field and electric potential. There have been some results on the number of critical points of the electric potential from knots, such as by Lipton (2021) and Lipton, Townsend, and Strogatz (2022). However, little analysis has been done on the electric field and electric potential using calculations for specific knots. We focus on torus knots, specifically a parametrization that embeds it on a torus centered at the origin with rotational symmetry about the z-axis. Particularly, in this project, we analyze the electric field along the z-axis to take advantage of symmetry. We also analyze the length of the knot as a simpler integral. We show that the electric field is zero only at the origin, and investigate the extreme points of the electric field and electric potential using numerical methods and calculations. We also demonstrate a new way to apply methods for contour integration in complex analysis to calculate the length, electric potential, and electric field, and provide an explicit approximation for the length of a torus knot.