2026/03/02 by Alexander R. Klotz · 1 voice
Mathematics · #math.GT
paper · pdf · doi:10.1088/1751-8121/ae862e
arxiv published 2026/03/02 · arxiv updated 2026/03/13
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≥α0 C3/4 for some constant α0. There is currently an order-of-magnitude range for the value of α0 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_TQQ>√(8π√(3))+(2π+√(2π+7√(3)-12) )Q-1/2≈6.60+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_TQQ<1.005⋅ 4π(5√(5)-8)/3≈13.39, 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.