Universality in the diffusion of knots

TL;DR

This study reveals a universal ratio between the diffusion constants of knotted ring polymers and linear polymers, determined by the ideal knot's crossing number, which remains constant across different polymer lengths.

Contribution

It introduces a universal ratio for diffusion constants of knotted versus linear polymers, linked to the ideal knot's crossing number, valid across various polymer models and lengths.

Findings

The diffusion constant ratio is constant with respect to the number of monomers.

The ratio is determined by the average crossing number of the ideal knot.

The ratio remains valid over a wide range of polymer lengths.

Abstract

We have evaluated a universal ratio between diffusion constants of the ring polymer with a given knot $K$ and a linear polymer with the same molecular weight in solution through the Brownian dynamics under hydrodynamic interaction. The ratio is found to be constant with respect to the number of monomers, $N$, and hence the estimate at some $N$ should be valid practically over a wide range of $N$ for various polymer models. Interestingly, the ratio is determined by the average crossing number ($N_{AC}$) of an ideal conformation of knotted curve $K$, i.e. that of the ideal knot. The $N_{AC}$ of ideal knots should therefore be fundamental in the dynamics of knots.