Presented here are some solutions of the boundary value problem for a closed ply having a circle as its contact line. Each strand is modeled as a twisted thin elastic rod with circular cross section. The contact force is normal to the tangents of the strands. The approximate shape of the thin ply is described by a system of ODEs of the 5-th order. Topologically the closed plies are subdivided into two groups: knots and links. The knots are formed by a single closed rod while the links are made up with two closed rods.
The numerical solutions were obtained by means of the parameter continuation using AUTO. Visualization is made with MAPLE. The color represents the magnitude of the contact force density for the 2-link, the tre- and pentafoil knots.

Link 1

This is an approximation of the exact solution related to the Villarceau circles. It is the only nontrivial linked/knotted analytical solution known. The rate of winding is constant. The contact force is zero.

Link 2

Trefoil knot

Pentafoil knot
MAIN

BACK