Knot types, homotopies and stability of closed elastic rods

Abstract

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines.