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Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds

Please always quote using this URN:urn:nbn:de:0296-matheon-320
  • We provide a number of new construction techniques for cubical complexes and cubical polytopes, and thus for cubifications (hexahedral mesh generation). As an application we obtain an instance of a cubical 4-polytope that has a non-orientable dual manifold (a Klein bottle). This confirms an existence conjecture of Hetyei (1995). More systematically, we prove that every normal crossing codimension one immersion of a compact 2-manifold into R3 is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubation of Boy's surface as a dual manifold immersion, and with an odd number of facets. Our explicit example has 17 718 vertices and 16 533 facets. Thus we get a parity changing operation for 3-dimensional cubical complexes (hexa meshes); this solves problems of Eppstein, Thurston, and others.

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Metadaten
Author:Alexander Schwartz, Günter M. Ziegler
URN:urn:nbn:de:0296-matheon-320
Referee:Martin Grötschel
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2004/02/01
Release Date:2004/02/01
Preprint Number:29
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