Role of flexibility in entanglement

Entanglement is essential to the function of many physical systems. Flexibility and length determine the extent to which the system can become entangled. Given a perfectly flexible unit-radius tube, several researchers have studied the minimum length needed to tie different types of knots. Can one obtain the same configurations with less flexible tubing? Does more flexibility always yield tighter knots? We demonstrate a phase change in flexibility beyond which more flexibility adds very little entanglement. This level of flexibility is surprisingly low and appears to have a global bound. Since tensile strength and flexibility act inversely, this level of flexibility provides the maximal tensile strength for materials that need to pack tightly. This is a basic design principle that should be observable in nature.

Figure 1

A tight Hopf link can be tied with a rope of flexibility $f=0.50$.

Figure 2

Minimum $L_f$ values for flexibilities between 0.01 and 1.00.

Figure 3

Minimum $L_f$ values for flexibilities between 0.30 and 1.00. Notice the phase change at 0.50.

Figure 4

The graphs of three of the knots with their fitting curves.

Figure 5

The graphs of the $4_1$ and $5_1$ knots cross near $f=0.20$.

Figure 6

The packed Hopf links exhibit a family of links for which arbitrarily little flexibility is sufficient to tie a ropelength-minimized conformation.

Figure 7

The amount of flexibility needed to tie a ropelength-minimized $(n,n+1)$ torus knot decreases as $n$ increases.

Figure 8

Minimal conformations of the $4_1$ and $5_1$ knots for various values of $f$.