## 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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