Four-thirds power law for knots and links

Abstract

Nature © Macmillan Publishers Ltd 1998 8 bounded above and below across each family. No p<3/4 can work, because CአA and there is a universal lower bound 12 for L/A 3/4 . Furthermore, rope-length minimizers in a family with L~C 3/4 must satisfy C~A; thus these tight curves have L~A 3/4 . Any knot or link arises from a 'braid' , a collection of N ascending arcs in a cylinder Z, joining N points on the bottom of Z to the same N points on the top. Z is bent into a solid torus S, the top and bottom disks are identified, and the arcs are joined to give a link in S. For a braid with bounds on its slope, curvature and horizontal strand separation, this bending fixes rope length within a uniform factor (depending on the shape of Z and S).For a family of N-component Hopf links with L~C 3/4 , we took a cylinder Z of height h and radius r. N points are distributed on the bottom disk of Z, and rotated one full turn as they ascend. The resulting braid (Fig. 1a) has N helical strands of bounded slope and curvature, provided h≈r; to separate strands, we needed r~N 1/2 . Thus the braid (and the corresponding link, Fig. 1b) has rope length L~hN~N 3/2 . As each component links every other component exactly once , CኑN(N-1); the standard projection has N(N-1)crossings, so C = N(N-1)~N 2 .For (N,N-1)-torus knots in S with LC 3/4 , we repeated this construction in the lower half of Z. To define the rest of the braid, we chose a circuit joining the N points (Fig. 1c) and slid each point along this circuit to the next while ascending through the upper half of Z. The horizontal distance a point travels is comparable to, at most, r. So, as before, we needed h~r~N 1/2 and rope length L~N 3/2 . The minimum crossing number of this knot 15 is C=N(N-2)~N 2 . (Lattice knots with the same growth of crossing number have been 10.

Opacity and entanglement of polymer chains

“…in perfect agreement with the exact results of [11,12] and with the numerical values for protein backbones.…”

Novel display of knotted DNA molecules by two-dimensional gel electrophoresis

“…1A), most knotted molecules (K) migrate faster than unknotted (C) or linear DNA (L). In these conditions, knot migration is known to be related to the average crossing number of the ideal representation of the corresponding knot (15)(16)(17)(18). In general, the higher the crossing number, the more compact the knot structure and the faster it migrates in the gel.…”

Knotting probability of DNA molecules confined in restricted volumes: DNA knotting in phage capsids

“…Because the electrophoretic migration of knots formed on the same-size DNA molecules is related to the average crossing number of the ideal representation of the corresponding knot (37) and this relation is approximately linear up to knots with 10 crossings (38), populations of knots with 3, 4, 5, 6, 7, 8, 9, and 10 crossings were assigned to the distinguishable gel bands in Fig. 2 A. Although it is known that some knots might deviate from this linear relation (39,40), a continuous correlation between electrophoretic migration and the number of knot crossings also applies to more complex knots (41). Therefore, as a first approximation we used linear extrap- olation to estimate the number of crossings of the more complex knot populations (Fig.…”