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

TL;DR

This paper investigates the ropelength of T(Q,Q) torus links, establishing bounds that are close to each other and demonstrating constructions that approach the theoretical lower bounds, advancing understanding of knot complexity.

Contribution

The work derives new bounds for the ropelength of T(Q,Q) torus links and constructs specific links that nearly achieve these bounds, improving the precision of ropelength estimates.

Findings

Upper bounds are within a factor of 1.77 of the lower bounds.

Derived a stronger lower bound based on convex hull analysis.

Constructed links that are within 6-60% of the theoretical lower bounds.

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.