TY - GEN A1 - Athanasiadis, Christos A. A1 - Rambau, Jörg A1 - Santos, Francisco T1 - The Generalized Baues Problem for Cyclic Polytopes II N2 - Given an affine surjection of polytopes $\pi: P \to Q$, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of $Q$ which are induced by the map $\pi$ has the homotopy type of a sphere. We extend earlier work of the last two authors on subdivisions of cyclic polytopes to give an affirmative answer to the problem for the natural surjections between cyclic polytopes $\pi: C(n,d') \to C(n,d)$ for all $1 \leq d < d' < n$. T3 - ZIB-Report - SC-98-43 KW - Generalized Baues Problem KW - Polyhedral Subdivisions KW - Induced Subdivisions KW - Poset KW - Spherical KW - Cyclic Polytopes KW - Bistellar Operations Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3862 ER - TY - GEN A1 - Rambau, Jörg A1 - Santos, Francisco T1 - The Generalized Baues Problem for Cyclic Polytopes I. N2 - The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first non-trivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertice for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many non-regular triangulations of cyclic polytopes. Beyond this, we find triangulations of $C(11,5)$ with flip deficiency. This proves---among other things---that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve. T3 - ZIB-Report - SC-98-14 KW - Generalized Baues Problem KW - Polyhedral Subdivisions KW - Induced Subdivisions KW - Poset KW - Spherical KW - Cyclic Polytopes KW - Bistellar Operations KW - Flip Defici Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3579 ER - TY - GEN A1 - Rambau, Jörg T1 - TOPCOM: Triangulations of Point Configurations and Oriented Matroids N2 - TOPCOM is a package for computing triangulations of point configurations and oriented matroids. For example, for a point configuration one can compute the chirotope, components of the flip graph of triangulations, enumerate all triangulations. The core algorithms implemented in TOPCOM are described, and implentation issues are discussed. T3 - ZIB-Report - 02-17 KW - triangulation KW - point configuration KW - oriented matroid KW - software KW - chirotope KW - circuit KW - cocircuit KW - symmetry KW - TOPCOM Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6849 ER - TY - GEN A1 - Rambau, Jörg T1 - Triangulierungen von Punktmengen und Polyedern N2 - Dieser Report wurde im Sommersemester 2000 an der TU Berlin in einer Spezialvorlesung über Triangulierungen von Punktmengen und Polyedern als Skriptum verwendet. Nach einem motivierenden Kapitel werden grundlegende Begriffe und Konstruktionen in der Theorie der Triangulierungen von Punktmengen und Polyedern vorgestellt. Danach werden als weiterführende Themen reguläre Triangulierungen, Sekundärpolytope, bistellare Operationen, höhere Stasheff-Tamari-Halbordnungen und Triangulierungen mit wenigen bzw. gar keinen Flips behandelt. Ein Kapitel über Enumeration und Optimierung beschließt die Zusammenstellung. T3 - ZIB-Report - 00-46 KW - point configuration KW - triangulation KW - polytope KW - polyhedron KW - Gale diagram KW - secondary polytope KW - cyclic polytope KW - graph of all triangulations KW - conn Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6145 ER - TY - GEN A1 - Pfeifle, Julian A1 - Rambau, Jörg T1 - Computing Triangulations Using Oriented Matroids N2 - Oriented matroids are combinatorial structures that encode the combinatorics of point configurations. The set of all triangulations of a point configuration depends only on its oriented matroid. We survey the most important ingredients necessary to exploit oriented matroids as a data structure for computing all triangulations of a point configuration, and report on experience with an implementation of these concepts in the software package TOPCOM. Next, we briefly overview the construction and an application of the secondary polytope of a point configuration, and calculate some examples illustrating how our tools were integrated into the {\sc polymake} framework. T3 - ZIB-Report - 02-02 KW - triangulation KW - oriented matroid KW - software KW - chirotope KW - circuit KW - cocircuit KW - symmetry KW - regular KW - secondary polytope KW - hypergeometric function Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6692 ER -