Ropelength-minimizing concentric helices and non-alternating torus knots

TL;DR

This work extends ropelength optimization from alternating to non-alternating torus knots by constructing multihelices—concentric shells of helices around a central rod—and optimizing both combinatorial distributions and geometric parameters. The authors derive tight asymptotic bounds for large systems, with a leading ropelength scaling of $L \sim 7.83\,Q^{3/2}$ and a precise prefactor, and show that torus closures into $T(pQ,Q)$ links can achieve ropelengths closely tracking the $C^{3/4}$ bound with substantially reduced coefficients (roughly $12$). They validate their designs through exhaustive combinatorial sweeps up to $N\le20$ (and case studies to $N=39$), interior infilling (reverse-Jenga) strategies, and verification with the Ridgerunner tightening tool. Collectively, the results reduce the previously known gap between upper and lower bounds for non-alternating torus knots from a factor of about 29 down to a range near $1.4$–$3.8$, representing a substantial tightening of ropelength bounds for this knot family. The work also outlines several avenues for future improvements, including relaxing geometric constraints, incorporating higher Fourier components, and exploring superhelical arrangements guided by differential-geometry concepts.

Abstract

An alternating torus knot or link may be constructed from a repeating double helix after connecting its two ends. A structure with additional helices may be closed to form a non-alternating torus knot or link. Previous work has optimized the dimensions and pitch of double helices to derive upper bounds on the ropelength of alternating torus knots, but non-alternating knots have not been studied extensively and are known to be tighter. Here, we examine concentric helices as units of non-alternating torus knots and discuss considerations for minimizing their contour length. By optimizing both the geometry and combinatorics of the helices, we find efficient configurations for systems with between 3 and 39 helices. Using insights from those cases, we develop an efficient construction for larger systems and show that concentric helices distributed between many shells have an optimized ropelength of approximately 7.83Q^(3/2) where Q is the total number of helices or the minor index of the torus knot, and the prefactor is exact and a 75 percent reduction from previous work. Links formed by extending these helices and bending them into a T(3Q,Q) torus link have a ropelength that is approximately 12 times the three-quarter power of the crossing number. These results reduce the ratio between the upper and lower bounds of the ropelength of non-alternating torus knots from 29 to between 1.4 and 3.8.

Figures (7)

• Figure 1: Side and top views of ideal cylindrical helices with 2, 3, 4, and 10 helices. The height and radius of the 10-helix are well approximated by the large-$N$ approximations.
• Figure 2: a. Artistic renderings of the Rod of Asclepius and the Caduceus, representing efficient 2- and 3-helices wiki. b. An ideal triple helix, and a shorter but wider double helix wound around a central rod. c. Improvement of Caduceus-style helices relative to an $N$-helix with the same topology. Above 5 helices, the improvement is well-described by Eq. \ref{['eq:cad']}. For 2 through 5 helices, the radius must be increased, reducing the improvement.
• Figure 3: Four geometrically optimized 12-multihelices with a rod and a varying distribution of helices between the first and second shells. The contour length is minimized for the 1-5-6 configuration. In all cases, the second shell is radially separated from the first by a distance of 2. The fifth image is the geometrically and combinatorially optimized 12-helix, which has a rod and three helices in a 1-4-5-2 arrangement at radii 2, 4, and 6.
• Figure 4: a. Ropelength per crossing of a winding of a 721-multihelix with a central rod and an equal number of helices per shell, as a function of the number of shells. The dependence is well-described by an upper bound derived from the sum of two limiting cases, Eq. \ref{['eq:upper']}. The horizontal lines show the improvements due to infilling the interior of the system, and further optimizing the arrangements by reverse-Jenga. b. Ropelength per crossing as a function of the total number of helices (including the rod). Green ticks show optimized configurations determined by exhaustive combinatirial sweeps and geometric minimization. Red data corresponds to an increment of 5 helices per shell, and the limiting trend from Eq. \ref{['eq:1089']}. Black data corresponds to an equal number of helices per shell, and the limiting trend from Eq. \ref{['eq:934']}. Blue data corresponds to infilling the equal-per-shell helices in increments of 4, and the limiting trend from Eq. \ref{['eq:784']}. Orange points show further improvements from a reverse-Jenga algorithm. All limiting trends depend inversely on the square root of the number of helices.
• Figure 5: a. Partial rendering of a $T(266,19)$ torus link with a 1-6-12 configuration. b. Rendering of a $T(48,16)$ torus link with an optimized 1-5-7-3 configuration. c. Ratio of ropelength to the three-quarter power of crossing number of $T(3Q,Q)$ torus links constructed from optimized helical configurations.