Ideal trefoil knot

Abstract

A set of self-contact points of the most tight, parametrically tied trefoil knot is determined. The knot is subjected to further tightening procedure based on the shrink-on-no-overlaps algorithm. Changes in the structure of the set of the self-contact points are monitored and the final form of the set is determined.

High resolution portrait of the ideal trefoil knot

“…Let us return to the analysis of the polygonal K p knot found by the SP-FEM simulation and consider the problem of the contact functions. Approximate discrete images of the functions have been presented in references [11], [14], [15] and [17]. As mentioned above, the SP-FEM algorithm analyzes the vertex-vertex distances (the vertices belonging to different parts of the simulated rope) and whenever they become smaller than 2, it introduces a pair of opposite forces aiming to remove the violations.…”

Orientational transition in nematic liquid crystals under oscillatory Poiseuille flow

“…For a steady flow along the x axis with a velocity field v x (z), v y = v z = 0 (rectilinear plane flow, typically of the Couette or Poiseuille type) the director will, in the absence of other torques, align in the flow plane (x − z plane) at the angle θ f l = ± arctg( α 3 /α 2 ) with the x axis if α 3 /α 2 > 0 (the ± sign corresponds to positive/negative shear rate ∂v x /∂z) [1,2]. In common nematics α 3 /α 2 is small but positive (≈ 0.01), however, in some materials (in particular near a nematic -smectic transition) one has α 3 /α 2 < 0 and instead of flow alignment there is more complicated tumbling motion [3,4,5,6,7]. Note that α 2 is always negative, so the sign of α 3 /α 2 is determined by that of α 3 .…”

The average shape of the closed trefoil knot fluctuating on a floppy rope

“…In what follows we shall discuss it using as an example the tightest of all trefoil knots tied on the perfect rope of radius R = 1 described parametrically by a set of three functions. Construction details of the polygonal knots K poly are described in previous publications [3]. Using the Rawdon method it is possible to inscribe a curvilinear knot into a polygonal knot [5].…”

“…There exists, however, a particular type of model rope for which the temperature of the thermal bath becomes irrelevant. This model rope, called perfect rope, being perfectly floppy is at the same time perfectly hard [3]. The rope is perfectly hard if the shapes of its perpendicular sections do not change even when the rope is squeezed.…”