Scaling behavior of knotted random polygons and self-avoiding polygons: Topological swelling with enhanced exponent

TL;DR

This paper investigates how the size of knotted self-avoiding polygons scales with the number of segments, revealing topological swelling and an enhanced scaling exponent, supported by numerical simulations.

Contribution

It demonstrates the topological swelling effect and the enhancement of the scaling exponent for knotted polygons, and introduces the additivity of equilibrium lengths for composite knots.

Findings

Knotted polygons are larger than unknotted ones when excluded volume is small.

The scaling exponent for polygons with fixed knots is enhanced.

The equilibrium length of composite knots is additive based on prime knots.

Abstract

We show that the average size of self-avoiding polygons (SAP) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We argue an "enhancement" of the scaling exponent for random polygons with a fixed knot. We study them systematically through SAP consisting of hard cylindrical segments with various different values of the radius of segments. Here we mean by the average size the mean-square radius of gyration. Furthermore, we show numerically that the equilibrium length of a composite knot is given by the sum of those of all constituent prime knots. Here we define the equilibrium length of a knot by such a number of segments that topological entropic repulsions are balanced with the knot complexity in the average size. The additivity suggests the local knot picture.