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Dual pairs of non-polytopal diagrams and spheres

Christoph Schulz
1985 Discrete Mathematics  
There is a dual pair of cellular decompositions of the 3-sphere both realizable as geometric complexes in Euclidean 4-space but neither isomorphic to the boundary complex of a convex 4-polytope.  ...  In the next section we shall construct a dual pair of non-polytopal 3diagrams which will be modified in the last section to yield a dual pair of non-polytopal 3-spheres in E4.  ...  A dual pair of non-polytopal 3-diagrams Another way of stating the Steinitz-problem is in terms of (d -1)-diagrams.  ... 
doi:10.1016/s0012-365x(85)80021-2 fatcat:tfhxel3h7na25gdrajudngui3e

Alexander duality in subdivisions of Lawrence polytopes

Francisco Santos, Bernd Sturmfels
2003 Advances in Geometry  
The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.  ...  Examples of Alexander dual pairs of subdivided Lawrence polytopes are given in Section 3. The smallest non-trivial example is the pair of triangular prisms in Figure 1 .  ...  Figure 2 . 2 This subdivision of a uniform Lawrence polytope (the triangular prism) is Alexander dual to a triangulation of a non-uniform Lawrence polytope (the pyramid).  ... 
doi:10.1515/advg.2003.012 fatcat:ji26u2okgrcynk56ishrjenlkm

Parametrization of Generalized Primal-Dual Triangulations [chapter]

Pooran Memari, Patrick Mullen, Mathieu Desbrun
2011 Proceedings of the 20th International Meshing Roundtable  
Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and  ...  Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular.  ...  Partial funding was generously provided by the National Science Foundation through grants CCF-1011944, CCF-0811373, and CMMI-0757106.  ... 
doi:10.1007/978-3-642-24734-7_13 dblp:conf/imr/MemariMD11 fatcat:pjbzq7fhebenvmfvj4x7slpnb4

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

Alexander Schwartz, Guenter M. Ziegler
2004 arXiv   pre-print
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).  ...  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.  ...  The illustrations of polytopes and balls were produced in the polymake system of Gawrilow & Joswig [14] , via javaview of Polthier et. al. [26] . (A small number of figures was drawn with xfig.)  ... 
arXiv:math/0310269v3 fatcat:dn2jsrowqzeejh6a7nrtfjqnnm

One-point suspensions and wreath products of polytopes and spheres

Michael Joswig, Frank H. Lutz
2005 Journal of combinatorial theory. Series A  
In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.  ...  Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties.  ...  The dual wedge of a polytope Let P ⊂ R d be a d-dimensional polytope (or d-polytope for short), and let v be a vertex.  ... 
doi:10.1016/j.jcta.2004.09.009 fatcat:ydm7rm7p4recbc2g2etn6kqkfe

Arbitrarily Large Neighborly Families of Congruent Symmetric Convex 3-Polytopes [chapter]

Scott Kim, Jeff Erickson
2003 Discrete Geometry  
a convex hull edge; the Schlegel diagram of the polar dual of any neighborly 4-polytope consists of a neighborly family of 3-polytopes.  ...  its degree-ordered sequence of non-zero coefficients.  ...  Figures 1 and 3 were produced with the help of the programs 'qhull' [2, 3] and 'geomview' [1, 19] .  ... 
doi:10.1201/9780203911211.ch19 fatcat:kl47afgoozddhbh6dl3rh7hxsy

One-Point Suspensions and Wreath Products of Polytopes and Spheres [article]

Michael Joswig, Frank H. Lutz
2004 arXiv   pre-print
In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.  ...  Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties.  ...  If K is a vertex-transitive non-spherical homology sphere and d ≥ 1, then ∂∆ d ≀ K is a non-PL sphere for d ≥ 1.  ... 
arXiv:math/0403494v1 fatcat:2lsx5bqwlbhbve3jo22muxk65e

Toric Topology. Chapter 2: Combinatorial structures [article]

Victor Buchstaber, Taras Panov
2012 arXiv   pre-print
Comments and suggestions are very welcome.  ...  Give an example of a non flag simple polytope with the nonnegative γ-vector. Alexander duality For any simplicial complex, a dual complex may be defined on the same set.  ...  The boundary of a simplicial n-polytope is an (n−1)-dimensional P L sphere. In dimensions n 3 every (n − 1)-diagram is a Schlegel diagram of an npolytope.  ... 
arXiv:1102.1081v2 fatcat:57zsd4dl6rgfzhjkvr3jyn5u2q

Small f-vectors of 3-spheres and of 4-polytopes [article]

Philip Brinkmann, Günter M. Ziegler
2016 arXiv   pre-print
In particular, we thus prove that there are f-vectors of cellular 3-spheres with the intersection property that are not f-vectors of any convex 4-polytopes, thus answering a question that may be traced  ...  By implementing this approach, we classify the f-vectors of 4-polytopes in the range f_0+f_3<22.  ...  Acknowledgements We are very grateful to Marge Bayer and Moritz Firsching valuable comments and discussions.  ... 
arXiv:1610.01028v1 fatcat:xnglibxgjzb5xop6x34hoaaa2i

Neighborly 4-polytopes with 9 vertices

Amos Altshuler, Leon Steinberg
1973 Journal of combinatorial theory. Series A  
A complete classification is given for neighborly 4-polytopes with 9 vertices. It is found that there are exactly 23 combinatorial types.  ...  In view of Theorem 1, and the remarks following Theorem 3, a necessary condition for the realizability of that 3-sphere as a 4-polytope, necessarily neighborly, is that the non-diagonal elements on each  ...  The determinant of the edge-valence matrix is an invariant of the combinatorial type qf the sphere, and is positive if the matrix is regular. Figure 2 .FIG. 2 . 22 Two pairs (NB, %?  ... 
doi:10.1016/0097-3165(73)90074-5 fatcat:xwg437px2vbwzniqozyemunoqi

The dunce hat in a minimal non-extendably collapsible 3-ball [article]

Bruno Benedetti, Frank H. Lutz
2013 arXiv   pre-print
As we will see, the Alexander dual of the dunce hat is collapsible.  ...  We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space.  ...  The triangulation D of the dunce hat is also contained in Grünbaum-Sreedharan's sphere No. 37 (polytopal, with 8 vertices and 20 facets) and in Grünbaum-Sreedharan's sphere M (non-polytopal, with 8 vertices  ... 
arXiv:0912.3723v2 fatcat:7n2rwt2chzbonmqcrutstleahm

k-set polytopes and order-k Delaunay diagrams

Dominique Schmitt
2006 2006 3rd International Symposium on Voronoi Diagrams in Science and Engineering  
k-set polytope of S, the order-k Voronoi diagram of S, and its dual, the order-k Delaunay diagram of S.  ...  of a (d + 1)dimensional k-set polytope and is the orthogonal dual of the order-k Voronoi diagram.  ...  a well known dual, the Voronoi diagram of S.  ... 
doi:10.1109/isvd.2006.43 dblp:conf/isvd/SchmittS06 fatcat:kg26avh6ivhn5m2lul4mcdqv7u

Oriented Matroids [chapter]

Jürgen Richter-Gebert, Günter Ziegler
2004 Handbook of Discrete and Computational Geometry, Second Edition  
Other introductions and basic sources of information include Bachem & Kern [1] , Bokowski [5] , Bokowski & Sturmfels [7] , and Ziegler [19, Lect. 6 and 7].  ...  We refer to the monograph by Björner, Las Vergnas, Sturmfels, White & Ziegler [3] for a broad introduction, and for an extensive development of the theory of oriented matroids with a reasonably complete  ...  construction of non-rational polytopes, as well as of non-polytopal spheres with d+4 vertices, etc.  ... 
doi:10.1201/9781420035315.ch6 fatcat:ic2fzbamuvgu7jrbp6y7o2ej4y

The triangulations of the 3-sphere with up to 8 vertices

David Barnette
1973 Journal of combinatorial theory. Series A  
The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined.  ...  Using similar methods we show that one cannot always preassign the shape of a facet of a 4-polytope.  ...  Figure 6 gives the Schlegel diagrams of the 9 combinatorial types of such polytopes. THEOREM 1.  ... 
doi:10.1016/0097-3165(73)90062-9 fatcat:5e4fqk3nbfeovoxodjeifhtyrm

Realizability and inscribability for simplicial polytopes via nonlinear optimization

Moritz Firsching
2017 Mathematical programming  
3-spheres with 10 vertices into polytopal and non-polytopal spheres.  ...  We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres.  ...  For the help in Section 3.2 I thank Frank Lutz, who provided the list of triangulations and David  ... 
doi:10.1007/s10107-017-1120-0 fatcat:fhkhmutqxbf37lryaio7bnotf4
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