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Linear Random Knots and Their Scaling Behavior Kenneth Millett, Akos Dobay,, and Andrzej Stasiak*,

Summary: Linear Random Knots and Their Scaling Behavior
Kenneth Millett, Akos Dobay,,§ and Andrzej Stasiak*,
Department of Mathematics, University of California, Santa Barbara, California 93106,
Laboratoire d'Analyse Ultrastructurale, Ba^timent de Biologie, UniversiteŽ de Lausanne,
1015 Dorigny, Switzerland, and Theoretische Physik, Fakultašt fušr Physik,
Ludwig-Maximilians-Universitašt, 80333 Mušnchen, Germany
Received June 20, 2004; Revised Manuscript Received October 21, 2004
ABSTRACT: We present here a nonbiased probabilistic method that allows us to consistently analyze
knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists
of analyzing the spectrum of knots formed by multiple closures of the same open walk through random
points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains
is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority
of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of
the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot
type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear
random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border
zone of the configuration space of two or more knot types. We also characterize the scaling behavior of
linear random knots.
Knots form on linear polymers and affect their physi-
cal behavior.1 However, the determination of the knot


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


Collections: Mathematics