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Energy, ropelength, and other physical aspects of equilateral knots

Summary: Energy, ropelength, and other physical aspects of
equilateral knots
Kenneth C. Millett a
, Eric J. Rawdon b,*,1
Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93016, USA
Department of Mathematics and Computer Science, Duquesne University, Pittsburgh, PA 15282, USA
Received 27 February 2001; received in revised form 29 October 2002; accepted 21 December 2002
Closed macromolecular chains may form physically knotted conformations whose relative occurrence and spatial
measurements provide insight into their properties and the mechanisms acting upon them. Under the assumption of a
degree of structural homogeneity, equilateral spatial polygons are a productive context within which to create math-
ematical models of these knots and to study their mathematical and physical properties. The ensembles, or spaces, of
these knots are models of the settings within which the knots evolve in ways determined by a physical model. In this
paper we describe the mathematical foundation of such models as well as such spatial, geometric, statistical, and
physical properties of the configurations as mathematical energies, thickness and ropelength, average crossing number,
average writhe, and volumes and surfaces areas of standard bodies enclosing the knots. We present methods with which
the energy and ropelength are optimized within the families of spatially equivalent equilateral configurations. Nu-
merical results from our implementation of these methods are shown to illustrate connections between the physical


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics