The Elastic Trefoil is the Doubly Covered Circle

To describe the behavior of knotted loops of springy wire with an elementary mathematical model we minimize the integral of squared curvature, E = ∫ ϰ 2 , together with a small multiple of ropelength R = length/thickness in order to penalize selfintersection. Our main objective is to characterize all limit configurations of energy minimizers of the total energy E ϑ ≡ E + ϑ R as ϑ tends to zero. For short, these limit configurations will be referred to as elastic knots . The elastic unknot turns out to be the once covered circle with squared curvature energy ( 2 π ) 2 . For all (non-trivial) knot classes for which the natural lower bound ( 4 π ) 2 on E is sharp, the respective elastic knot is the doubly covered circle. We also derive a new characterization of two-bridge torus knots in terms of E , proving that the only knot classes for which the lower bound ( 4 π ) 2 on E is sharp are the ( 2 , b ) -torus knots for odd b with | b | ≥ 3 (containing the trefoil knot class). In particular, the elastic trefoil knot is the doubly covered circle.