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~$E_{\mathrm{bend}}=\int\kappa^2$ and add a small multiple of ropelength~${\mathcal R}=\textnormal{length}/\textnormal{thickness}$ in order to penalize selfintersection. Our main objective is to characterize {\it elastic knots}, i.e., all limit configurations of energy minimizers of the total energy $E_{\vartheta}:=E_{\mathrm{bend}}+\vartheta{\mathcal R}$ as $\vartheta$ tends to zero. For every odd $b>1$ and the respective class of $(2,b)$-torus knots (containing the trefoil) we obtain a complete picture showing that the respective elastic $(2,b)$-torus knot is the twice covered circle.