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

A unit-vector field n on a convex three-dimensional polyhedron (P) over bar 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) over bar is given. The classification is determined by-certain invariants, namely edge orientations (values of n on the edges of (P) over bar), kink numbers (relative winding numbers of n between edges on the faces of (P) over bar), and wrapping numbers (relative degrees of n on surfaces separating the vertices of (P) over bar), 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.