# Knotting and weak knotting in confined, open random walks using virtual knots

## TL;DR

This study investigates knotting in open, confined polymers using virtual knots, revealing a prevalent weak knotting behavior that depends on confinement but is largely independent of chain length.

## Contribution

It introduces virtual knots as a tool to classify ambiguous knotting in open polymers and compares knotting behaviors across different confinement geometries.

## Key findings

- Weak knotting probability increases with chain length in confined polymers.
- Weak knotting is strongly correlated with confinement degree, not chain length.
- Unconfined polymers show minimal weak knotting across studied lengths.

## Abstract

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves. Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.