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

## TL;DR

This paper establishes a lower bound on the number of critical points of the electric potential around a knot, linking it to the knot's tunneling number and using Morse and stable manifold theories.

## Contribution

It introduces a novel lower bound on critical points of the electric potential based on the knot's tunneling number, connecting knot theory with potential theory.

## Key findings

- Number of critical points ≥ 2 * tunneling number + 2
- Uses Morse theory and stable manifold theory for proof
- Provides a new link between knot invariants and potential critical points

## 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.