Tight Bounds for Tight Links: Ropelength of T(Q,Q) torus links

Abstract

Ropelength, L, is a parameter characterizing the minimum contour length of a knot or link. There exist upper and lower bounds on ropelength with respect to crossing number, C, including a universal lower bound constraining $L\geqα_0 C^{3/4}$ for some constant $α_0$. There is currently an order-of-magnitude range for the value of $α_0$ between 1.105 and 10.76. In this work, we show that T(Q,Q) torus links can be constructed such that the upper bound is within a factor of 1.77 of the lower bound. We derive a stronger lower bound based on the convex hull around close-packed disks of approximately $α_{T_{QQ}}>\sqrt{8π\sqrt{3}}+(2π+\sqrt{2π+7\sqrt{3}-12}\ )Q^{-1/2}\approx6.60+7.61Q^{-1/2}$, significantly higher than the best universal lower bound of 1.105. We show that a link can be constructed without any free parameters or geometric optimization that, when $Q$ is large, has a coefficient $α_{T_{QQ}}<1.005\cdot 4π(5\sqrt{5}-8)/3\approx13.39$, and can be improved to to 11.68 by solving a helical no-overlap constraint equation. For $Q$ up to 20 we construct links from smooth planar curves or toroidal helices minimized with respect to a small number of geometric parameters, that are between 6 and 60% greater in ropelength than the lower bound. Many such links can be annealed to within 10% of the lower bound using gradient descent. This represents significant progress towards developing sharp bounds on the ropelengths of specific classes of knots and links.

Figures (5)

• Figure 1: a. Diagram of a $T(7,7)$ link. b. The shape of the shortest curve that can be part of a unit-thickness $T(7,7)$ link, with its bold centerline pushed a distance of 1 from the minimal convex hull around 6 unit disks. The lower bound arises from assuming that each component of a $T(Q,Q)$ link is the minimal convex hull of $Q-1$ unit disks. c. Diagram of a $T(92,92)$ link. d. An estimate of the minimal hull around 91 disks. For large enough $Q$ the minimal hull approximates a circular curve around a hexagonal packing.
• Figure 2: Construction of an efficient $T(Q,Q)$ link from concentric helices. a. The arrangement of helices, incrementing 4 per shell. b. The vertical concentric helices. c. A torus formed from wrapping the helix, shown with thinner curves. d. Two copies of the torus are increased in major radius and linked through each other, shown with unit thickness curves. The visible extra space highlights the unconstrained nature of the helices.
• Figure 3: a. Three-quarter coefficients for the $T(Q,Q)$ links formed by doubly-linked tori of concentric helices incremented by 4 and 5 per shell, as well as the optimal construction. The two curves in each color show the best- and worst-case scenarios, which depend on how filled the outer helices are. Kinks correspond to configurations below which the major radius must be increased to allow for double torus linking. Limiting values are shown as dashed lines without the helical correction and the lower bound is shown for comparison. b. Shows the same data as a ratio relative to the lower bound.
• Figure 4: a. Three-quarter coefficients for the best constructed $T(Q,Q)$ links for $Q$ from 2 to 20, color-coded by their construction method. Values reached by gradient descent are shown in red and the lower bound is shown for comparison. The dashed line corresponds to the best large-$Q$ upper bound of 11.68. The red datum at $Q=3$ is from Ashton et al. ashton2011knot. b. The same data as a ratio relative to the lower bound. The solid black line below the data indicates the ratio for the gradient descent data assuming that numerically optimized convex hulls minimal, and the green line indicates the ratio for the constructed links assuming the isoperimetric inequality. The large-$Q$ ratio of 1.77 sets the upper limit of this plot.
• Figure 5: a. Typical shape of a gibbous curve that can be used to construct links. b-d. Optimal links for $Q=3,4,5$ constructed from gibbous curves, shown with reduced tube radius. d. includes a rounded square as the central curve. e-g. $T(20,20)$ links formed by optizing a 20-helix (e), congruent circles (f), and gibbous curves (g), shown with unit tube radius.