Symmetric critical knots for O'Hara's energies

TL;DR

This paper proves the existence of symmetric critical torus knots for O'Hara's energies within a specific parameter range, confirming numerical observations and revealing multiple critical knots per class.

Contribution

It establishes the existence of symmetric critical torus knots for O'Hara's energies using symmetric criticality principles, a novel application in knot energy theory.

Findings

At least two smooth critical knots per torus knot class.

Supports numerical experiments with theoretical proof.

Focuses on energies with parameter α in (2,3).

Abstract

We prove the existence of symmetric critical torus knots for O'Hara's knot energy family $E_\alpha$, $\alpha\in (2,3)$ using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth $E_\alpha$-critical knots, which supports experimental observations using numerical gradient flows.