On the Topology and Geometric Construction of Oriented Matroids and Convex Polytopes

Richter, J.; Sturmfels, Bernd

doi:10.2307/2001676

Cited by 3 publications

(9 citation statements)

References 10 publications

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“…Since Finschi and Fukuda developed a database of oriented matroids [21,22] containing non-uniform ones, the realizability classification of larger oriented matroids including non-uniform case has begun. Existing non-realizability certificates such as non-Euclideanness [18,35] and biquadratic final polynomials [9] and existing realizability certificates such as non-isolated elements [44] and solvability sequence [12] were applied to OM (4,8) and OM (3,9) [24,39,40]. A new realizability certificate using polynomial optimization and generalized mutation graphs [40] and new non-realizability certificates non-HK* [24] and applying semidefinite programming [36] were proposed and applied to OM (4,8) and OM (3,9).…”

Section: Brief History Of Related Enumerationmentioning

confidence: 99%

Complete Enumeration of Small Realizable Oriented Matroids

Fukuda

Miyata

Moriyama

2012

Discrete Comput Geom

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Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases.Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e., non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points and 5dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices.

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Section: Brief History Of Related Enumerationmentioning

confidence: 99%

Complete Enumeration of Small Realizable Oriented Matroids

Fukuda

Miyata

Moriyama

2012

Discrete Comput Geom

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“…There are uniform neighborly oriented matroids without universal edges in OM (5,11), OM (5,12), OM (7,11) and OM(9, 12) (only one such example, in OM(5, 10), was known [19]). The latter (together with their realizability) gives a positive answer to a question by Richter and Sturmfels [50] concerning the existence of neighborly 2k-polytopes with 2k + 4 vertices without universal edges.…”

Section: Introductionmentioning

confidence: 82%

“…We can certify realizability for those polytopes obtained by sewing and Gale-sewing [46], and we have non-realizability certificates by biquadratic final polynomials [17]. Moreover, certain cases can be decided by studying their universal edges [50]. In particular, we are able to completely classify: 5. all possible combinatorial types of neighborly 8-polytopes with 12 vertices.…”

Section: Introductionmentioning

confidence: 99%

Enumerating Neighborly Polytopes and Oriented Matroids

Miyata

Padrol

2015

Experimental Mathematics

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Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures.In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids -a combinatorial abstraction of neighborly polytopes -of small rank and corank. In particular, if we denote by OM(n, r) the set of all oriented matroids of rank r and n elements, we determine all uniform neighborly oriented matroids in OM(5, ≤ 12), OM(6, ≤ 9), OM(7, ≤ 11) and OM(9, ≤ 12) and all possible face lattices of neighborly oriented matroids in OM(6, 10) and OM (8,11). Moreover, we classify all possible face lattices of uniform 2-neighborly oriented matroids in OM(7, 10) and OM (8,11). Based on the enumeration, we construct many interesting examples and test open conjectures. arXiv:1408.0688v2 [math.CO]

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“…Some of the maps in Figure 1 are also used to study the slack realization space in the matroid setting [BW19]. In fact, essentially all the results in this paper have a direct translation to the case of matroid realization spaces, which is still an active area of research [RS91], [FMM13].…”

Section: Introductionmentioning

confidence: 91%

“…However, there are still many spheres for which it is not known whether they are polytopal or not, see e.g., [CS19, Section 5] and [Zhe20, Remark 5.2]. For the matroid case similar realizability issues arise, as one can see in [RS91,FMM13].…”

Section: The Reduced Slack Modelmentioning

confidence: 99%

Combining realization space models of polytopes

Gouveia¹,

Macchia²,

Wiebe³

2020

Preprint

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In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of [NZ19, Problem 6.1], about the realizability of a family of simplicial spheres, in general in the negative by proving the non-realizability of one of their simplicial spheres.

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On the Topology and Geometric Construction of Oriented Matroids and Convex Polytopes

Cited by 3 publications

References 10 publications

Complete Enumeration of Small Realizable Oriented Matroids

Complete Enumeration of Small Realizable Oriented Matroids

Enumerating Neighborly Polytopes and Oriented Matroids

Combining realization space models of polytopes

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