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Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds | | Alexander Schwartz
; Guenter M. Ziegler
; | | Date: |
17 Oct 2003 | | Subject: | Combinatorics MSC-class: 52B12; 52B11; 52B05; 57Q05 | math.CO | | Abstract: | 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 R^3 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. | | Source: | arXiv, math.CO/0310269 | | Services: | Forum | Review | PDF | Favorites | |
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