Homotopy classification of knotted defects in bounded domains

TL;DR

This paper extends the homotopy classification of knotted defects in ordered media from three-dimensional space to handlebodies, allowing defects to reach boundaries and describing them via monodromies and planar diagrams.

Contribution

It introduces a new classification scheme for defects in handlebodies, incorporating boundary-reaching defects through monodromies and diagrammatic representations.

Findings

Classification scheme for handlebody defects using monodromies.

Application to octahedral frame fields and biaxial nematic liquid crystals.

Examples demonstrating the extended classification method.

Abstract

Nozaki et.~al.\ gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence relation. We extend their result by giving a classification scheme for ordered media in handlebodies, where defects are allowed to reach the boundary. Through monodromies around meridional loops, global defects are described in terms of planar diagrams whose edges are colored by elements of the fundamental group of the order parameter space. We exhibit examples of this classification in octahedral frame fields and biaxial nematic liquid crystals.