Universal properties of knotted polymer rings

TL;DR

This study uses Monte Carlo simulations to investigate the universal properties of knotted polymer rings across different lattices, confirming some universal behaviors in their entropic, metric, and geometrical features.

Contribution

It provides the first comprehensive analysis of universality in knotted polymer rings across multiple lattice types, including asymptotic knot probability ratios and geometrical entanglement measures.

Findings

Knot probability ratios are lattice-independent at large N.

Mean squared radius of gyration scaling depends only on correction to scaling.

Writhe distribution's standard deviation exhibits a universal power-law behavior.

Abstract

By performing Monte Carlo sampling of $N$-steps self-avoiding polygons embedded on different Bravais lattices we explore the robustness of universality in the entropic, metric and geometrical properties of knotted polymer rings. In particular, by simulating polygons with $N$ up to $10^5$ we furnish a sharp estimate of the asymptotic values of the knot probability ratios and show their independence on the lattice type. This universal feature was previously suggested although with different estimates of the asymptotic values. In addition we show that the scaling behavior of the mean squared radius of gyration of polygons depends on their knot type only through its correction to scaling. Finally, as a measure of the geometrical self-entanglement of the SAPs we consider the standard deviation of the writhe distribution and estimate its power-law behavior in the large $N$ limit. The estimates of the power exponent do depend neither on the lattice nor on the knot type, strongly supporting an extension of the universality property to some features of the geometrical entanglement.