Measuring the entanglement complexity of 3-periodic networks through the untangling number

TL;DR

This paper introduces a novel measure called the untangling number to quantify the entanglement complexity of 3-periodic networks, which are models for crystalline structures, using knot theory to identify least tangled configurations.

Contribution

It defines ground states of 3-periodic networks via knot diagrams and introduces the untangling number as a new metric for entanglement complexity.

Findings

Ground states are identified using knot-theoretic crossing diagrams.

The untangling number quantifies the minimal entanglement of a network.

This measure links network configuration to physical properties.

Abstract

Periodic networks serve as models for the structural organisation of biological and chemical crystalline systems. Single or multiple networks can have different configurations in space, where entanglement may arise due to the way the (possibly curvilinear) edges weave around each other. This entanglement influences the functional, physical, and chemical properties of the materials modelled by the networks, which highlights the need to quantify its complexity. In this paper, we define the least tangled embeddings of 3-periodic networks that we call ground states, through the use of knot-theoretic crossing diagrams. The concept of a ground state permits the definition of a measure of entanglement complexity called the untangling number that quantifies the distance between a given 3-periodic structure and its least tangled version.