Abundance of unknots in various models of polymer loops

TL;DR

This study investigates the prevalence of unknots in different polymer loop models, revealing how segment length distributions influence the transition from unknotted to knotted states.

Contribution

It provides a computational analysis of how the characteristic length scale for knotting varies across polymer models with different segment length distributions.

Findings

N_0 is smallest (~240) for uniform segment lengths

N_0 is larger (~305) for Gaussian segments

N_0 can be very large with fat-tailed segment distributions

Abstract

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $ \sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.