The polymer theta-point as a knot delocalisation transition

TL;DR

This paper investigates how prime flat knots in self-attracting polymers transition from localized to delocalized states across different temperature regimes, revealing a weak localization at the collapse point.

Contribution

It provides numerical evidence for knot delocalization at the polymer collapse transition and proposes conjectures for exact exponents based on polymer network theory.

Findings

Knots are localized in the high temperature swollen phase.

Knots become delocalized in the low temperature globular phase.

At the collapse transition, knots are weakly localized.

Abstract

We study numerically the tightness of prime flat knots in a model of self-attracting polymers with excluded volume. We find that these knots are localised in the high temperature swollen regime, but become delocalised in the low temperature globular phase. Precisely at the collapse transition, the knots are weakly localised. Some of our results can be interpreted in terms of the theory of polymer networks, which allows to conjecture exact exponents for the knot length probability distributions.