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Construction techniques for cubical complexes, odd cubical 4-polytopes, and prescribed dual manifolds. (English) Zbl 1110.52015

Summary: 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 nonorientable 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 \(\mathbb R^3\) is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. As an instance we obtain a cubical 4-polytope with a cubification 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 three-dimensional cubical complexes (hex meshes); this solves problems of Eppstein, Thurston, and others.

MSC:

52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B11 \(n\)-dimensional polytopes
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
57Q05 General topology of complexes

Software:

polymake; JavaView
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