The elastic trefoil is 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.

47 pages, 28 figures; minor corrections

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