Length of the tightest trefoil knot

The physical sense of tight knots provided by the SONO algorithm is discussed. A method allowing one to predict their length is presented. An upper bound for the minimum length of a smooth trefoil knot is determined.

Figure 1

When two pieces of the rope pass too close to each other, they overlap.

Figure 2

When a bend in the rope is too sharp, its surface develops defects.

Figure 3

Corrugated model of the perfect rope.

Figure 4

The tightest trefoil knot tied on the corrugated rope consisting of $n=60$ cells.

Figure 5

Single cell of the corrugated rope.

Figure 6

The shape of the corrugated rope cell in the situation of maximal bending.

Figure 7

Bending angle at the $i\mathrm{th}$ vertex.

Figure 8

The tightest turn of the corrugated rope.

Figure 9

During the tightening move of Eq. (10), each vertex is moved by a small amount toward the center of its osculating circle.

Figure 10

Top: the polygonal $K_p$ and inscribed $K_c$ knots defined by the set of vertices delivered by the knot tightening algorithm. Bottom: the smooth inscribed knot $K_c$ tied with the the perfect rope of radius $R_c$.

Figure 11

The raw polygonal $L_p$, inscribed $L_c$, and weighted average $L_a$ ropelengths of the tightest SONO trefoil knots tied on the corrugated rope versus the number of vertices.

Figure 12

A tight conformation of the (59,2) torus knot.

Figure 13

The tightest torus wound around a straight rope; the smooth and corrugated cases.

Figure 14

Position of one of the spheres, of which the corrugated torus is constructed, is controlled by the angle $\phi$. In the picture, the angle has its maximum value ${\phi }_{\max }$.

Figure 15

Geometry of the $n$-gon skeleton of the corrugated torus.

Figure 16

Polygonal, inscribed, and weighted average ropelengths of the tightest trefoil knots tied on the corrugated rope versus the inverse of the number of vertices. Curves shown in the figure were obtained by fitting the data with the function given by (31).