Warren, I don't seem to have received the original message, at least not recently. Can you please state what is is you are responding to? Thanks, Dan << just use Euler characteristic x=V-E+F for polyhedra with V vertices E edges F faces. For genus g x=2-2g=#handles. For polyhedra of type "sphere" x=2. (example cube: V=8, E=12, F=6). For polyhedra of type "torus" x=0. For polyhedra of type "K spheres" we have x=2K. For polyhedra of type "sets of surfaces" the Euler characteristics obviously sum. So this whole problem becomes totally trivial if you view in terms of Euler. The end.

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