Equilibria of elastic cable knots and links
E.L. Starostin & G.H.M. van der Heijden
We present a theory for equilibria of geometrically exact braids made of two
thin, uniform, homogeneous, isotropic, initially-straight, inextensible and
unshearable 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
bifurcation diagram and selected numerical solutions illustrate our approach.
keywords: elastic knots and links, cable knots, equilibria, variational
problem, bifurcation
in: New Directions in Geometric and Applied Knot Theory, S. Blatt,
Ph. Reiter, A. Schikorra (eds) (De Gruyter, Berlin/Boston, 2018), pp. 258-275