# On the regularity of critical points for O'Hara's knot energies: From smoothness to analyticity

## TL;DR

This paper proves that smooth critical points of certain knot energies are actually analytic, extending previous results and using advanced complex analysis techniques to establish higher regularity.

## Contribution

It demonstrates the analyticity of critical points for O'Hara's knot energies with specific parameters, advancing understanding of their regularity properties.

## Key findings

- Critical points are analytic under given conditions.
- Extension of regularity results from smoothness to analyticity.
- Method based on Cauchy's majorants and gradient decomposition.

## Abstract

We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{\alpha,p}$, with $p=1$ and $2<\alpha< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy critical points of $\mathcal{E}^{\alpha,1}$ subject to a fixed length constraint are not only $C^\infty$ but also analytic. Our approach is based on Cauchy's method of majorants and a decomposition of the gradient that was adapted from the M\"obius energy case $\mathcal{E}^{2,1}$ in \cite{BV19}.