Discrete Geometry Group
| Discrete Geometry Group | |
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Technische Universität Berlin
Fakultät II: Institut für Mathematik, MA 6-2,
Straße des 17. Juni 136
10623 Berlin
Germany
| Room: | MA 621 |
| Phone: | +49-30-314-25753 |
| Fax: | +49-30-314-21269 |
| Email: | witte"AT"math.tu-berlin.de |
Member of the Research Group Polyhedral Surfaces
Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special attention is paid to the cube case.
Explicit construction (and electronic model) of a construction by M. Hachimori and G. M. Ziegler [Math. Zeitsch. 235: 159-171, 2000].
Explicit construction (and electronic model) of a construction by M. Hachimori and G. M. Ziegler [Math. Zeitsch. 235: 159-171, 2000].
Explicit construction (and electronic model) of a construction by M. Hachimori and G. M. Ziegler [Math. Zeitsch. 235: 159-171, 2000].
Explicit construction (and electronic model) of a construction by M. Hachimori and G. M. Ziegler [Math. Zeitsch. 235: 159-171, 2000].
Explicit construction (and electronic model) of a 3-ball which contains a knotted spanning arc consisting of one edge only.
Explicit construction (and electronic model) of a counterexample to the Perles conjecture by C. Haase and G. M. Ziegler [Disc. and Comp. Geom. 28: 29-44, 2002].
This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake.
There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \Omega(nd-1) vertices. For d=3 this was known, with examples provided by the "Ukrainian easter eggs" by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2.
Branched covers are applied frequently in topology - most prominently in the construction of
closed oriented PL d-manifolds. In particular, strong bounds for the number
of sheets and the topology of the branching set are known for dimension d<=4.
On the other hand, Izmestiev and Joswig described how to obtain a simplicial covering space (the
Here is a movie of how to build the trefoil as odd subcomplex via stellar subdivision of eight edges of the cyclic 4-polytope on seven vertices.
Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave a combinatorial equivalent of the Hilden and Montesinos result, constructing closed oriented combinatorial 3-manifolds as simplicial branched covers of combinatorial 3-spheres. The construction of Izmestiev and Joswig is generalized and applied to the result of Piergallini, obtaining closed oriented combinatorial 4-manifolds as simplicial branched covers of simplicial 4-spheres.
This is a concise summery of the first three chapters of my dissertation thesis. However, the results presented in "Constructing Simplicial Branched Covers" are stronger and the proofs shorter.
For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary.