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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.… (More)
There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has Ω(n d−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.
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.
Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. Special attention is paid to the cube case. Such triangulations are instrumental in deriving lower bunds for the number of real roots of certain sparse polynomial systems by recent results of… (More)
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects.
The software project polymake  has been developed since 1997 in the Discrete Geometry group at TU Berlin by Ewgenij Gawrilow and Michael Joswig, with contributions by several others. It was initially designed to work with convex polytopes. Due to its open design the polymake framework can also be used on other types of objects; the current release… (More)
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… (More)
Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d ≤ 4 every closed oriented PL… (More)