Global curvature, thickness, and the ideal shapes of knots

Abstract

The global radius of curvature of a space curve is introduced. This function is related to, but distinct from, the standard local radius of curvature and is connected to various physically appealing properties of a curve. In particular, the global radius of curvature function provides a concise characterization of the thickness of a curve, and of certain ideal shapes of knots as have been investigated within the context of DNA.

Structural motifs of biomolecules

“…The self-avoidance of an ensemble of hard spheres, each of radius ⌬, can be ensured by considering all pairs of spheres and requiring that none of the distances between the sphere centers is Ͻ2⌬. The self-avoidance of a tube of thickness ⌬ can be enforced through a suitable three-body potential (11,12). We denote the tube axis by a smooth curve, r(s), where the arc-length s satisfies 0 Յ s Յ L, and L is the total length of the tube.…”

“…We begin by describing the discrete version of a tube of thickness ⌬ represented by a chain of coins, whose planes coincide with equally spaced circular cross-sections of the tube. The self-avoidance of the tube is implemented by the three-body prescription (11,12) described in the legend of Fig. 1.…”

Universal growth law for knot energy of Faddeev type in general dimensions

“…The geometric and physical contents of a knot may be measured by an energy functional, E, which is non-negative valued and sometimes assumed to be scale invariant, whose choice unambiguously reflects one's standpoint on what properties of a knot are to be taken into account. Wellknown knot energies designed for measuring knotted/tangled space curves include the Gromov distortion energy (Gromov 1978(Gromov , 1983, the Möbius energy (O'Hara 1991(O'Hara , 1992Bryson et al 1993;Freedman et al 1994) and the ropelength energy (Nabutovsky 1995;Buck 1998;Cantarella et al, 1998Cantarella et al, , 2002Gonzalez & Maddocks 1999). See Janse van Rensburg (2005) for a rather comprehensive survey of these and other knot energies and related interesting studies.…”

Following Knots down Their Energy Gradients

“…It will be of value to persons who wish to perform similar experiments of their own. In order to give the reader access to some of this work we have particularly listed references [3][4][5][6][7][8][9][10][11][12][13][14] as a partial sample of work in this field.…”