Global Defect Topology in Nematic Liquid Crystals

TL;DR

This paper provides a comprehensive topological classification of nematic liquid crystal textures with defects, linking mathematical invariants to physical configurations and experimental observations.

Contribution

It introduces a global homotopy classification framework for nematic textures with defects using twisted cohomology and explicit computations for knotted defects.

Findings

Homotopy classes correspond to the first homology group of a branched double cover.

Planar textures represent classes of order 2, characterized by merons.

Relation established between global topology and experimental phenomena like $ au$ lines.

Abstract

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in $\mathbb{R}^3$, showing that the distinct homotopy classes have a 1-1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group have representatives that are planar and characterise the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterisation. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of $\tau$ lines in cholesterics.