The elastic trefoil is the twice covered circle

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.