Equilibria of elastic cable knots and links

We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homogeneous, isotropic, initially-straight, inextensible and unshear- able elastic rods of circular cross-section. We formulate a second-order variational problem for an action functional whose Euler–Lagrange equations, partly in Euler– Poincaré form, yield a compact system of ODEs for which we define boundary-value problems for braids closed into knots or links. The purpose of the chapter is to present a pathway of deformations leading to braids with a knotted axis, thereby offering a way to systematically compute elastic cable knots and links. A representative bifurca- tion diagram and selected numerical solutions illustrate our approach.