What is the length of a knot in a polymer?

TL;DR

This paper introduces statistical methods to define and measure the length of knots in polymer rings, revealing how knot length scales with polymer size in different solvent conditions through Monte Carlo simulations.

Contribution

It provides a novel statistical framework for knot length measurement and demonstrates how knot localization varies with solvent quality using simulation data.

Findings

Knot length scales as N^0.75 in good solvent conditions

Knot delocalizes with length proportional to N in collapsed regime

Monte Carlo results support the scaling laws for different regimes

Abstract

We give statistical definitions of the length, l, of a loose prime knot tied into a long, fluctuating ring macromolecule. Monte Carlo results for the equilibrium, good solvent regime show that < l > ~ N^t, where N is the ring length and t ~ 0.75 is independent of the knot topology. In the collapsed regime below the theta temperature, length determinations based on the entropic competition of different knots within the same ring show delocalization (t~1).