  Unit-vector fields $\nvec$ on a convex polyhedron $P$ subject to tangent
boundary conditions provide a simple model of nematic liquid crystals in
prototype bistable displays. The equilibrium and metastable configurations
correspond to minimisers and local minimisers of the Dirichlet energy, and may
be regarded as $S^2$-valued harmonic maps on $P$. We consider unit-vector
fields which are continuous away from the vertices of $P$.
  A lower bound for the infimum Dirichlet energy for a given homotopy class is
obtained as a sum of minimal connections between fractional defects at the
vertices of $P$. In certain cases, this lower bound can be improved by
incorporating certain nonabelian homotopy invariants. For a rectangular prism,
upper bounds for the infimum Dirichlet energy are obtained from locally
conformal solutions of the Euler-Lagrange equations, with the ratio of the
upper and lower bounds bounded independently of homotopy type. However, since
the homotopy classes are not weakly closed, the infimum may not be realised;
the existence and regularity properties of continuous local minimisers of given
homotopy type are open questions. Numerical results suggest that some homotopy
classes always contain smooth minimisers, while others may or may not depending
on the geometry of $P$. Numerical results modelling a bistable device suggest
that the observed nematic configurations may be distinguished topologically.
