Complex asymptotics of the M\"obius energy gradient of symmetric helix pairs

TL;DR

This paper analyzes the asymptotic behavior of the M"obius energy gradient for symmetric helix pairs, revealing divergence patterns and implications for stationary helicoids as the coiling ratio increases.

Contribution

It characterizes the limiting behavior of the M"obius energy gradient for symmetric helix pairs using complex asymptotics, providing new insights into their equilibrium states.

Findings

Gradient diverges depending on radius relative to 1/2

Stationary helicoids tend to radius 1/2 as coiling increases

Results have implications for M"obius-Plateau energy and minimal surfaces

Abstract

The M\"obius energy is a well-studied knot energy with nice regularity and self-repulsive properties. Stationary curves under the M\"obius energy gradient are of significant theoretical interest as they they can indicate equilibrium states of a curve under its own forces. In this paper, we consider stationary symmetric helix pairs under the M\"obius energy. Through methods of complex asymptotics, we characterize the limiting behavior of the M\"obius gradient as the coiling ratio tends to infinity: the gradient will diverge in opposing directions depending on whether the radius is less than or greater than $\frac{1}{2}$. We conclude by discussing the implications to the more general M\"obius-Plateau energy, where the energy of a curve, or pair of curves, includes the area of the minimal surface bounded by them. Symmetric helix pairs bound a helicoid between them, and applying our result shows that stationary helicoids grow to radius $\frac{1}{2}$ from below as the coiling tends to infinity.