Ropelength of superhelices and (2, n)-torus knots

Abstract

In this paper we investigate a ropelength-minimizing conformation of 4-strand superhelical strings whose axial curves constitute the standard double helix.

In IIuh et al (2016 J. Phys. A: Math. Theor. 49 415205) the authors found a specific conformation of standard double helix which was mathematically shown to be the unique ropelength-minimizing conformation. Adopting the conformation as axial curves we present a parametrization of superhelical curves so that the resulting shape is controlled by the twist number N and the helical radius r(2). For each N, the value of r(2) minimizing the ropelength is numerically estimated. A notable observation is that the ropelength per crossing of our 4-strand superhelical strings is minimized around 2.96 < N < 2.97 which suggests a moment of transition from tightly-packed status to unpacked status.

As an application of our estimation, we derive an upper bound on the ropelength of (2, 6k + 1)-torus knots, which is 45.8237k + 28.4223. Finally the efficiency of our superhelix model for (2, n)-torus knots is discussed in comparison with the circular helix model.