Percolation and Dissolution of Borromean Networks

TL;DR

This paper investigates the percolation threshold and dissolution properties of Borromean networks inspired by DNA experiments, revealing a slightly higher percolation threshold than standard lattices and differences in topological link release upon dissolution.

Contribution

It introduces a lattice model for Borromean networks, analyzes their percolation threshold, and compares dissolution properties to other linked networks, providing insights for synthetic chemistry applications.

Findings

Percolation threshold at approximately 60.75% occupancy.

Dissolution yields more isolated loops than Hopf-linked networks.

Simulation results can predict experimental outcomes.

Abstract

Inspired by experiments on topologically linked DNA networks, we consider the connectivity of Borromean networks, in which no two rings share a pairwise-link, but groups of three rings form inseparable triplets. Specifically, we focus on square lattices at which each node is embedded a loop which forms a Borromean link with pairs of its nearest neighbors. By mapping the Borromean link network onto a lattice representation, we investigate the percolation threshold of these networks, (the fraction of occupied nodes required for a giant component), as well as the dissolution properties: the spectrum of topological links that would be released if the network were dissolved to varying degrees. We find that the percolation threshold of the Borromean square lattice occurs when approximately 60.75\% of nodes are occupied, slightly higher than the 59.27\% typical of a square lattice. Compared to the dissolution of Hopf-linked networks, a dissolved Borromean network will yield more isolated loops, and fewer isolated triplets per single loop. Our simulation results may be used to predict experiments from Borromean structures produced by synthetic chemistry.