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 $L\geq\alpha_0 C^{3/4}$ for some constant $\alpha_0$. There is currently an order-of-magnitude range for the value of $\alpha_0$ 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 $\alpha_{T_{QQ}}>\sqrt{8\pi\sqrt{3}}+(2\pi+\sqrt{2\pi+7\sqrt{3}-12}\ )Q^{-1/2}\approx6.60+7.61Q^{-1/2}$, 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 $\alpha_{T_{QQ}}<1.005\cdot 4\pi(5\sqrt{5}-8)/3\approx13.39$, and can be improved to to 11.68 by solving a helical no-overlap constraint equation that requires a conjectural approximation. 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.