The elastic trefoil is the twice covered circle

TL;DR

This paper studies the minimal elastic energy configurations of knotted loops of wire, revealing that for certain knot classes, the elastic knot is a twice covered circle, especially the trefoil knot.

Contribution

It characterizes elastic knots as limits of energy minimizers, showing that the elastic trefoil is a twice covered circle, which is a novel insight into knot elasticity.

Findings

Elastic unknot is a round circle with energy (2π)^2.

For certain knot classes, the elastic knot is a twice covered circle.

The elastic trefoil is identified as the twice covered circle.

Abstract

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\int\kappa^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic knots, i.e., all limit configurations of energy minimizers of the total energy $E_\vartheta:=E_{\text{bend}}+\vartheta\mathcal R$ as $\vartheta$ tends to zero. The elastic unknot turns out to be the round circle with bending energy $(2\pi)^2$. For all (non-trivial) knot classes for which the natural lower bound $(4\pi)^2$ for the bending energy is sharp, the respective elastic knot is the twice covered circle. The knot classes for which $(4\pi)^2$ is sharp are precisely the $(2,b)$-torus knots for odd $b$ with $|b|\ge 3$ (containing the trefoil). In particular, the elastic trefoil is the twice covered circle.