Topological correlations in trivial knots: new arguments in support of the crumpled polymer globule

TL;DR

This paper demonstrates that in a collapsed phase, unknotted polymer rings have a fractal, crumpled structure with topological correlations indicating that parts of the chain are almost trivial knots, supporting the crumpled globule model.

Contribution

It provides analytical and numerical evidence for the fractal crumpled structure of collapsed unknotted polymers and characterizes topological correlations within dense knots.

Findings

Knot complexity scales linearly with strip length for long strips.

Parts of trivial knots tend to be almost unknotted with complexity proportional to the square root of length.

Topological state of any part of a collapsed trivial knot is nearly trivial.

Abstract

We prove the fractal crumpled structure of collapsed unknotted polymer ring. In this state the polymer chain forms a system of densely packed folds, mutually separated in all scales. The proof is based on the numerical and analytical investigation of topological correlations in randomly generated dense knots on strips $L_{v} \times L_{h}$ of widths $L_{v}=3,5$. We have analyzed the conditional probability of the fact that a part of an unknotted chain is also almost unknotted. The complexity of dense knots and quasi--knots is characterized by the power $n$ of the Jones--Kauffman polynomial invariant. It is shown, that for long strips $L_{h} \gg L_{v}$ the knot complexity $n$ is proportional to the length of the strip $L_{h}$. At the same time, the typical complexity of the quasi--knot which is a part of trivial knot behaves as $n\sim \sqrt{L_{h}}$ and hence is significantly smaller. Obtained results show that topological state of any part of the trivial knot in a collapsed phase is almost trivial.