  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.
