The Existence of Energy Minimizers for Knots and Links

Abstract

In this paper, I will introduce the mathematical ideas of knots and links, define a function on each that we call the energy of a knot or link, and prove several lemmas and theorems that relate the knot and link energies to basic concepts in knot theory. The main results of this paper will be the proofs of two theorems on the existence of energy minimizers for both knots and links. Finally, I will compare the depth and structure of these two major proofs in order to better understand the nature of the results.