Elastic knots

May 22, 2023
To model knotted springy wires we minimize in prescribed knot classes a total energy consisting of the classic Euler-Bernoulli bending energy and an additive repulsive potential. The ultimate goal is to characterize the shape of such minimizing knots for various knot classes. For that we send a prefactor of the repulsive potential to zero and analyze the limiting configurations -- so-called elastic knots. For all torus knot classes T(2,b) we established the doubly-covered circle as the unique elastic knot, which confirms mechanical and numerical experiments. There are, however, instances when numerical gradient flows seem to get stuck in different configurations exhibiting some symmetry. To provide analytic support for these rare observations we use the symmetric criticality principle to find symmetric elastic knots exhibiting these symmetries. In this talk we give a survey on the analytic results, show some of the numerical simulations obtained by Bartels, Riege and Reiter, and address many open questions.