The average crossing number of equilateral random polygons

Abstract

In this paper, we study the average crossing number of equilateral random walks and polygons. We show that the mean average crossing number ACN of all equilateral random walks of length n is of the form 3 16 •n•ln n+O(n). A similar result holds for equilateral random polygons. These results are confirmed by our numerical studies. Furthermore, our numerical studies indicate that when random polygons of length n are divided into individual knot types, the ACN(K) for each knot type K can be described by a function of the form ACN(K) = a • (n − n 0) • ln(n − n 0) + b • (n − n 0) + c where a, b and c are constants depending on K and n 0 is the minimal number of segments required to form K. The ACN(K) profiles diverge from each other, with more complex knots showing higher ACN(K) than less complex knots. Moreover, the ACN(K) profiles intersect with the ACN profile of all closed walks. These points of intersection define the equilibrium length of K, i.e., the chain length ne(K) at which a statistical ensemble of configurations with given knot type K-upon cutting, equilibration and reclosure to a new knot type K-does not show a tendency to increase or decrease ACN(K). This concept of equilibrium length seems to be universal, and applies also to other length-dependent observables for random knots, such as the mean radius of gyration Rg .

The spectrum of filament entanglement complexity and an entanglement phase transition

“…Thousand steps, edges have no width, so no self-avoidance. For equilateral flights in this case, it has been proven that the ACN ∼ L log L (Diao et al 2003). For both sorts of random flights, it has been proven that the entanglement is at least approximately L (Sumners & Whittington 1988).…”

“…The patterns are ideal objects-the filaments have no length or other physical characteristics. Polymer theory brings a different perspective, studying the properties of various models of large collections of randomly arranged filaments (De Gennes 1979;Grosberg & Khokhlov 1994;van Rensburg 2000). Complex properties of polymeric materials and fluids have been modelled, predicted and quantified with considerable accuracy.…”

The Generation of Random Equilateral Polygons

“…The four methods discussed here have been applied throughout the literature to make estimations of many potentially interesting measures of the shape of a polygon, including the radius of gyration, the average crossing number, the diameter, the miniball radius, the box and skinny box dimensions (as well as the associated surface area and volume), the volume and surface area of the convex hull of the polygon, the inertial asphericity, and the enveloping ellipsoid asphericity [11,17,18,20,21,39,59,62,63,66,67]. Another application of particular importance for this study is the estimation of the total curvature and total torsion of the polygon [60].…”

“…Polymer chains have been modeled as freely jointed random polygons under certain conditions [10,15,18,21,29,55,60,62,63,66]. This simple representation of polymeric chains derives from their statistical properties under theta conditions, in which the effects of excluded volume have been eliminated [9,10,26].…”

“…For example, one might be interested in how the average behavior of circular polymers of a certain knot type compares with that of another knot type, or with the average properties of the entire population of polymers. In this setting, there is also a growing body of experimental [1-3, 22, 23, 64, 70] and numerical research [14,15,18,21,29,50,55,60,62,63,67].…”