Knot Tightening by Constrained Gradient Descent

Abstract

ABSTRACT. We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent.We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the polyhedral cone of variations of a polygon of thickness one which do not decrease thickness to first order is finitely generated, and give an explicit set of generators. Using this cone we give a first-order minimization procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength criticality.Our main numerical contribution is a set of 379 almost-critical knots and links, including all prime knots with ten and fewer crossings and all prime links with nine and fewer crossings. For links, these are the first published ropelength figures, and for knots they improve on existing figures. We give new maps of the self-contacts of these knots and links, and discover some highly symmetric tight knots with particularly simple looking self-contact maps.

Length of excitable knots

“…The centerline of the tube is an ideal knot if the tube has minimal length within a given knot topology and this length is the rope length of the knot [10]. Extensive data on the computed rope lengths of knots can be found in [18]. As rope length is defined as the ratio of length to diameter, then to fix a normalization we need to define the diameter of our excitable knots.…”

How nice are critical knots? Regularity theory for knot energies

“…Despite of the lack of an explicit analytical characterization of the shape of a (non-trivial) tight knot, there are several contributions to discretization and numerical visualization, see [Ashton et al, 2011;Cantarella et al, 2005;Carlen & Gerlach, 2012;Carlen et al, 2005;Gerlach, 2010;Gonzalez et al, 2002b;Smutny, 2004]. An interesting packing problem, namely maximizing length for prescribed thickness on the two-dimensional sphere S 2 , has been tackled by Gerlach and von der Mosel [2011a;.…”

“…Diao, Ernst, and Janse van Rensburg [1999] considered a slightly different notion on C 1 -curves. See Ashton et al [2011] for further references. Gonzalez and Maddocks [1999] provided an alternative characterization-not requiring any initial regularity-by the minimum value of the circumradius function R(γ(s), γ(t), γ(u)) over all triplets of points of γ. Here…”

On the groundstate energy spectrum of magnetic knots and links

“…The minimum energy spectra of the first prime knots and links is determined by setting h = 0 in (13) and by using the ropelength data (λ K ) obtained by the RIDGERUNNER tightening algorithm [1] for each knot/link type K. A particularly simple expression is obtained by normalizing m(λ K , 0) with respect to the minimum energy value m • of the tight torus; thus, we havem…”