Discrete & Computational Geometry

, Volume 43, Issue 2, pp 242–262

Hamiltonian Submanifolds of Regular Polytopes


DOI: 10.1007/s00454-009-9151-9

Cite this article as:
Effenberger, F. & Kühnel, W. Discrete Comput Geom (2010) 43: 242. doi:10.1007/s00454-009-9151-9


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.


Hamiltonian subcomplex Centrally-symmetric Tight PL-taut Intersection form Pinched surface Sphere products 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für Geometrie und TopologieUniversität StuttgartStuttgartGermany