Classification of unit-vector fields in convex polyhedra with tangent boundary conditions

TL;DR

This paper provides a homotopy classification of tangent unit-vector fields in convex polyhedra, using invariants like edge orientations, kink numbers, and wrapping numbers, with applications to liquid crystal physics.

Contribution

It introduces a comprehensive classification framework for tangent unit-vector fields in convex polyhedra based on topological invariants, extending previous understanding.

Findings

Classification determined by edge orientations, kink numbers, and wrapping numbers.

Sum rules constrain the invariants, ensuring consistent classifications.

Application to liquid crystal configurations in polyhedral cells.

Abstract

A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The classification is determined by certain invariants, namely edge orientations (values of n on the edges of P), kink numbers (relative winding numbers of n between edges on the faces of P), and wrapping numbers (relative degrees of n on surfaces separating the vertices of P), which are subject to certain sum rules. Another invariant, the trapped area, is expressed in terms of these. One motivation for this study comes from liquid crystal physics; tangent unit-vector fields describe the orientation of liquid crystals in certain polyhedral cells.