Symmetric critical knots for O'Hara's energies

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.

30 pages, streamlined proofs of Lemma 4.6 and Lemma A.1, typos corrected