Borromean Hypergraph Formation in Dense Random Rectangles

TL;DR

This paper investigates how dense packings of randomly oriented rectangles can form Borromean links and hypergraphs, revealing percolation thresholds and implications for topological entanglement in biological systems.

Contribution

It introduces a minimal stochastic model for Borromean link formation in dense rectangle packings without using knot invariants, highlighting percolation phenomena.

Findings

Dense packings can form Borromean triplets and clusters.

Percolation of Borromean hypergraphs occurs at specific densities.

Implications for topological connectivity in kinetoplast DNA.

Abstract

We develop a minimal system to study the stochastic formation of Borromean links within topologically entangled networks without requiring the use of knot invariants. Borromean linkages may form in entangled solutions of open polymer chains or in Olympic gel systems such as kinetoplast DNA, but it is challenging to investigate this due to the difficulty of computing three-body link invariants. Here, we investigate randomly oriented rectangles densely packed within a volume, and evaluate them for Hopf linking and Borromean link formation. We show that dense packings of rectangles can form Borromean triplets and larger clusters, and that in high enough density the combination of Hopf and Borromean linking can create a percolating hypergraph through the network. We present data for the percolation threshold of Borromean hypergraphs, and discuss implications for the existence of Borromean connectivity within kinetoplast DNA.