Ensnarled: On the topological linkage of spatially embedded network pairs

TL;DR

This paper introduces a graph theoretical method to analyze and quantify the topological entanglement of spatially embedded network pairs, focusing on their linkage and un-linking properties using Hopf-link identification.

Contribution

It develops a novel edge priority operator and a greedy algorithm to identify critical edges for unlinking, providing a new topological metric for ensnarled networks.

Findings

The method successfully identifies critical edges in ensnarled networks.

The algorithm optimizes edge removals for un-linking.

A new topological metric characterizes the ensnarled state.

Abstract

The observation, design and analysis of mesh-like networks in bionics, polymer physics and biological systems has brought forward an extensive catalog of fascinating structures of which a subgroup share a particular, yet critically under appreciated attribute: being embedded in space such that one wouldn't be able to pull them apart without prior removal of a subset of edges, a state which we here call ensnarled. In this study we elaborate on a graph theoretical method to analyze ensnarled finite, 2-component nets on the basis of Hopf-link identification. Doing so we are able to construct an edge priority operator, derived from the linking numbers of the spatial graphs' cycle bases, which highlights critical edges. On its basis we developed a greedy algorithm which identifies optimal edge removals to achieve unlinking, allowing for the establishment of a new topological metric characterizing the state of ensnarled network pairs.