Monte Carlo Results for Projected Self-Avoiding Polygons: A Two-dimensional Model for Knotted Polymers

TL;DR

This paper introduces a 2D lattice model for knotted polymers, using a generalized algorithm to study their statistics, revealing a transition from self-avoiding walk to branched polymer behavior as crossing fugacity increases.

Contribution

It presents a novel 2D lattice model and a generalized algorithm for analyzing knotted polymers, including topological transformations, with numerical results on knot localization and phase transition behaviors.

Findings

Knot localization occurs at low crossing fugacity.

Increasing crossing fugacity induces a transition to branched polymer behavior.

Numerical evidence supports the ergodicity of the algorithm within fixed knot types.

Abstract

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two- dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favor of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: In the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot. Increasing the crossing fugacity gives rise to a transition from a self-avoiding walk to a branched polymer behavior.