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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 51; no. 48; pp. 485203 - 485215
• Main Authors: Huh, Youngsik, Kim, Hyoungjun, Oh, Seungsang
• Format: Journal Article
• Language: English
• Published: IOP Publishing 30.11.2018
• Subjects: ropelength; supercoil; superhelix; torus knot
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aae969

In this paper we investigate a ropelength-minimizing conformation of 4-strand superhelical strings whose axial curves constitute the standard double helix. In Huh 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 r2. For each N, the value of r2 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 -torus knots, which is 45.8237k  +  28.4223. Finally the efficiency of our superhelix model for -torus knots is discussed in comparison with the circular helix model.