Matched asymptotic expansions for twisted elastic knots: A self-contact problem with non-trivial contact topology

We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist, and held at both endpoints at infinity. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically nonlinear. The problem is formulated as a nonlinear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. We obtain a family of equilibrium solutions depending on a single loading parameter over(U, -) (proportional to applied twisting moment divided by square root of pulling force), which are asymptotically valid in the limit of a loose knot, ε{lunate} → 0. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium.