Ropelength of Tight Polygonal Knots

Abstract

Abstract. A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.

Biarcs, Global Radius of Curvature, and the Computation of Ideal Knot Shapes

“…We note that pt is bounded below by half of the two-point Euclidean distance function, i.e. by pp(s, σ) := 1 2 |q(s) − q(σ)|. At double critical points, and therefore at any off-diagonal global minimiser of pt, pp(s, σ) = pt(s, σ), but the global minimum of pp is always zero, and is achieved along the diagonal for any curve.…”

“…More precisely when we denote by dc the set of arguments (s, σ) ∈ I × I with s = σ that satisfy (1), then 17 ∆[q] = min min s∈I ρ(s), 1 2 min (s,t)∈dc |q(s) − q(t)| ,…”

New energy and helicity bounds for knotted and braided magnetic fields

“…By using ropelength data from knot tightening simulations obtained by SONO relaxation algorithm (Pieranski 1998, Baranska et al 2005 and, from (23), by letting M Ã ð0Þ ¼ 4=3 =2 2=3 , we obtain the groundstate energy spectrum of the first 250 prime knots. Let us normalize the energy levels with respect to the minimum energy value M Ã of the tight torus; we haveM The energy spectrum ofM for the first 250 prime knots, plotted for increasing ropelength within each family of minimum number of crossing, is shown in figure 6.…”

On the groundstate energy of tight knots

“…Technical details of this algorithm are widely available in the literature; the interested reader can consult several papers such as Pieranski et al (2001), Pieranski & Przybyl (2002) and Baranska et al (2004Baranska et al ( , 2005. We present here the ideas behind the algorithm and we highlight some critical aspects.…”

“…The SONO algorithm is a numerical software implemented by Przybyl (2001) under the direction of Pieranski (1998), and subsequently improved by other collaborators. Technical details of this algorithm are widely available in the literature; the interested reader can consult several papers such as Pieranski et al (2001), Pieranski & Przybyl (2002) and Baranska et al (2004Baranska et al ( , 2005. We present here the ideas behind the algorithm and we highlight some critical aspects.…”