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arXiv · 1908.01942

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

Max Lipton
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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 $\Phi(x) = \int_0^{2\pi} \frac{|r'(t)|}{|x - r(t)|}dt$. 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.

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Max Lipton. 2019-08-06. A lower bound on critical points of the electric potential of a knot. https://arxiv.org/abs/1908.01942

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