A statistical approach to knot confinement via persistent homology

TL;DR

This paper introduces a topological method using persistent homology to analyze how random knots occupy space, revealing correlations between geometric features and topological invariants as knot length varies.

Contribution

It presents a novel statistical framework combining persistent homology with geometric analysis to study knot confinement and deviations from ideal configurations.

Findings

Correlations increase with knot length.

Topological features relate to geometric quantities.

Framework detects deviations from ideal knots.

Abstract

In this paper we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology of the Vietoris-Rips complexes built from point clouds associated to knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated to the embedding and persistent homology based features, as a function of the knots' lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots.