Geometrical complexity of conformations of ring polymers under topological constraints

TL;DR

This paper investigates how topological constraints influence the geometrical complexity of ring polymers, revealing that such constraints reduce crossing numbers especially in thinner polymers, and that complexity and size are independent measures.

Contribution

It provides numerical analysis showing the impact of topological constraints on the geometrical complexity of ring polymers and distinguishes complexity from size as independent conformational measures.

Findings

Topological constraints decrease average crossing numbers for large N.

Thickness significantly affects the reduction in geometrical complexity.

Crossing number and size are independent conformational measures.

Abstract

One measure of geometrical complexity of a spatial curve is the number of crossings in a planar projection of the curve. For $N$-noded ring polymers with a fixed knot type, we evaluate numerically the average of the crossing number over some directions. We find that the average crossing number under the topological constraint are less than that of no topological constraint for large $N$. The decrease of the geometrical complexity is significant when the thickness of polymers is small. The simulation with or without a topological constraint also shows that the average crossing number and the average size of ring polymers are independent measures of conformational complexity.