We prove that the neighborly cubical polytopes studied by Günter M. Ziegler and the first named author  arise as a special case of the neighborly cubical spheres constructed by Babson, Billera and Chan . By relating the two constructions we obtain an explicit description of a non-polytopal neighborly cubical sphere and, further, a new proof of the fact that the cubical equivelar surfaces of McMullen, Schulz and Wills  can be embedded into ℝ3.
KeywordsSimplicial Complex Vector Representation Face Lattice Boundary Sphere Cubical Complex
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- D. Gale, Neighborly and cyclic polytopes, in Proceedings of Symposia in Pure Mathematics, Vol. VII, American Mathematical Society, Providence, R.I., 1963, pp. 225–232.Google Scholar
- B. Grünbaum, Convex Polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003, Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler.Google Scholar
- F. H. Lutz, The Manifold Page, 1999–2005, http://www.math.tu-berlin.de/diskregeom/stellar/.
- K. Polthier, K. Hildebrandt, E. Preuss and U. Reitebuch, JavaView v3.03, www.javaview.de, 2004.Google Scholar
- J. Richter-Gebert, Realization Spaces of Polytopes, Lecture Notes in Mathematics, vol. 1643, Springer-Verlag, Berlin, 1996.Google Scholar
- R. Sanyal, T. Rörig and G. M. Ziegler, Polytopes and Polyhedral Surfaces via Projection, in preparation.Google Scholar
- T. Rörig, Neighborly cubical spheres and polytopes, Master’s thesis, Technische Universität Berlin, 2004, http://www.math.tu-berlin.de/~thilosch/.
- G. M. Ziegler, personal communication, March 2004.Google Scholar