Tightening elastic (n, 2)-torus knots

E.L. Starostin & G.H.M. van der Heijden

We present a theory for equilibria of elastic torus knots made of a single thin, uniform, homogeneous, isotropic, inextensible, unshearable rod of circular cross-section. The theory is formulated as a special case of an elastic theory of geometrically exact braids consisting of two rods winding around each other while remaining at constant distance. We introduce braid strains in terms of which we formulate a second-order variational problem for an action functional that is the sum of the rod elastic energies and constraint terms related to the inextensibility of the rods. The Euler-Lagrange equations for this problem, partly in Euler-Poincaré form, yield a compact system of ODEs suitable for numerical solution. By solving an appropriate boundary-value problem for these equations we study knot equilibria as the dimensionless ropelength parameter is varied. We are particularly interested in the approach of the purely geometrical ideal (tightest) limit. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. For the pentafoil we find a symmetry-breaking bifurcation.

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