Knot probabilities in equilateral random polygons

TL;DR

This study investigates the probability of knotting in equilateral random polygons, revealing a universal scaling law involving exponential and power law factors, with improved data generation and knot detection methods.

Contribution

Introduces a universal scaling formula for knotting probability in equilateral polygons, with enhanced algorithms for data generation and knot detection.

Findings

Knotting probability follows a universal scaling law.

Unknot probability scales with a small negative power law.

Method improvements lead to more accurate knot detection.

Abstract

We consider the probability of knotting in equilateral random polygons in Euclidean 3-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.