  For piecewise-linear maps the stable and unstable manifolds of hyperbolic
periodic solutions are themselves piecewise-linear. Hence compact subsets of
these manifolds can be represented using polytopes (i.e. polygons, in the case
of two-dimensional manifolds). Such representations are efficient and exact so
for computational purposes are superior to representations that use a large
number of points on some mesh (as is usually done in the smooth setting). We
introduce a method for computing convex polytope representations of stable and
unstable manifolds. For an unstable manifold we iterate a suitably small subset
of the local unstable manifold and prior to each iteration subdivide polytopes
where they intersect the switching manifold of the map. We prove the output
converges to the (entire) unstable manifold and use it to visualise attractors
and bifurcations of the three-dimensional border-collision normal form: we
identify a heterodimensional-cycle, a two-dimensional unstable manifold whose
closure appears to be a unique attractor, and a piecewise-linear analogue of a
first homoclinic tangency where an attractor appears to be destroyed.
