A lower bound on critical points of the electric potential of a knot

Abstract

Given a knot $K$ parametrized by $r:[0,2\pi ]\to {\mathbb{R}}^3$, we can define the electric potential on its complement by $\Phi (x)={\int }_0^{2\pi }\frac{|r^{\prime }(t)|}{|x-r(t)|}dt$. Physicists and knot theorists want to understand the critical points of the potential and their behavior. The tunneling number $t(K)$ of a knot is the smallest number of arcs one needs to add to a knot so the complement is a handlebody. We show the number of critical points of the potential is at least $2t(K)+2$. The result is proven using Morse theory and stable manifold theory.