Hamiltonian Submanifolds of Regular Polytopes
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- Effenberger, F. & Kühnel, W. Discrete Comput Geom (2010) 43: 242. doi:10.1007/s00454-009-9151-9
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We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k-Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S2×S2. By this example all regular cases of n vertices with n<20 or, equivalently, all cases of regular d-polytopes with d≤9 are now decided.