Under-knotted and over-knotted polymers: compact self-avoiding loops

TL;DR

This study uses computer simulations to analyze the geometry and topology of compact self-avoiding loops, revealing insights into how knots behave and are distributed along compact polymers.

Contribution

It introduces a simulation approach to study the topological and geometric properties of compact polymers, highlighting the potential de-localization of knots in such systems.

Findings

Short under-knotted loops are more crumpled than average.

Knot de-localization may occur in compact polymers.

Global topological constraints influence local geometry.

Abstract

We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The importance of ergodic sampling of all loops is emphasized. We first look at the effect of global topological constraint on the local fractal geometry of a typical loop. We find that even short pieces of a compact trivial knot, or some other under-knotted loop, are somewhat crumpled compared to topology-blind average over all loops. We further attempt to examine whether knots are localized or de-localized along the chain when chain is compact. For this, we perform computational decimation and chain coarsening, and look at the "renormalization trajectories" in the space of knots frequencies. Although not completely conclusive, our results are not inconsistent with the idea that knots become de-localized when the polymer is compact.