Knot probabilities in equilateral random polygons. (9th September 2021)

• Record Type: Journal Article
• Authors: Xiong, A; Taylor, A J; Dennis, M R; Whittington, S G
• Is Part Of: Journal of physics. Volume 54:Number 40(2021)
• Publication Date: 2021-09-09
• Subjects: polygons -- knotting -- random walks -- knots; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ac1fc2
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We consider the probability of knotting in equilateral random polygons in Euclidean three-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous studies that indicated a purely exponential scaling. The methodology incorporates several improvements over previous investigations: our random polygon data set is generated using a fast, unbiased algorithm, and knotting is detected using an optimised set of knot invariants based on the Alexander polynomial.