Knotted Nematics

TL;DR

This paper uses knot theory to analyze knotted nematic textures, providing explicit constructions and linking number assignments, which enhances understanding of complex defect arrangements in nematic materials.

Contribution

It introduces a knot-theoretic framework for characterizing knotted nematics, including explicit constructions for knots with Milnor fibrations and boundary conditions.

Findings

Explicit closed-form constructions for knotted nematics.

Method to assign linking numbers to nematic links.

Relevance to experimental observations of Hopf links.

Abstract

Knotted line defects in continuous fields entrain a complex arrangement of the material sur- rounding them. Recent experimental realisations in optics, fluids and nematic liquid crystals make it important to fully characterise these textures and to understand how their properties relate to the knot type. We characterise knotted nematics through an application of classical knot theory founded upon the Pontryagin-Thom construction for nematic textures and give explicit closed form constructions for knots possessing Milnor fibrations with general boundary conditions. For links we construct nematic textures corresponding to all possible assigments of linking numbers and discuss the relevance to recent, and classic, experiments on Hopf links.