Gaps in the numbers of vertices of cubical polytopes, I

Blind, G.; Blind, R.

doi:10.1007/bf02574013

Cited by 17 publications

(17 citation statements)

References 5 publications

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“…Some torsion linear conditions have appeared among the linear conditions for f-vectors of certain cubical complexes [1,2,7]. However, the complete answer is not yet known for these complexes, whereas in our 2-strata case we have a complete answer.…”

Section: Introductionmentioning

confidence: 58%

Eulerian 2-Strata Spaces

Chen

Yan

1999

Journal of Combinatorial Theory, Series A

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In (B. F. Chen and M. Yan, 1998, Eulerian stratification of polyhedra, Adv. in Appl. Math. 21, 22 57), we introduced the notion of Eulerian stratified spaces. In this paper, we study in detail the topological and combinatorial properties of Eulerian 2-strata spaces, as the first step toward a deeper understanding of more strata cases. We show that Eulerian 2-strata spaces have similar topological structure as topological 2-strata spaces. Moreover, we find all the linear conditions on f-vectors over arbitrary abelian group. Academic Press

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Section: Introductionmentioning

confidence: 58%

Eulerian 2-Strata Spaces

Chen

Yan

1999

Journal of Combinatorial Theory, Series A

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“…Let i := y − k − 2 and j := y(y−3) (6) it follows that y ≥ 0 and from (7) it follows that k ≥ 0 if i ≥ 2. If i = 1 and j = 0, then (y, k) = (3, 0).…”

Section: Edge and Ridge Numbers Of 4-polytopesmentioning

confidence: 99%

“…(viii) The possible vertex numbers of cubical 3-polytopes are Π 0 (F 3 cub ) = {8} ∪ {n ∈ Z : n ≥ 10}.Blind & Blind[7] proved that the number of vertices f 0 as well as of edges f 1 are even for every cubical d-polytope if d ≥ 4 is even. According toBlind & Blind [8, Cor.…”

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confidence: 99%

Semi-algebraic sets of f-vectors

Sjöberg

Ziegler

2019

Isr. J. Math.

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Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f -vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such semi-algebraic sets of lattice points have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections.We develop proof techniques in order to show that, despite the observations above, some f -vector sets are not semi-algebraic sets of lattice points. This is then proved for the set of all pairs (f 1 , f 2 ) of 4-dimensional polytopes, the set of all f -vectors of simplicial d-polytopes for d ≥ 6, and the set of all f -vectors of general d-polytopes for d ≥ 6. For the f -vector set of all 4-polytopes this remains open.

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“…We deal with topological cubical complexes ~ in ~d having additional properties, namely: Such a topological cubical complex is called a special topological cubical (d -t)complex or (d -1)-STC-complex. Clearly, the boundary complex of a cubical d-polytope with fewer than 2 d § 1 vertices is a (d -1)-STC-complex ( (6) is Lemma 3.5, and (7) is satisfied for every cubical polytope, see [3]).…”

Section: Topological Cubical Complexes and Stc-complexesmentioning

confidence: 99%

“…For cubical polytopes it is known that fj(P) > fy(C), where C is a combinatorial cube [2]. In [3] it is shown that fo(P) is even for all even d > 4, and it is reasonable to expect that more gaps occur, especially for numbers slightly larger than fo(C) = 2 d. We determine here all the gaps up to 2 a § that is, we determine all the possible fo(P) up to 2 d+l. However, we do much more.…”

Section: Introductionmentioning

confidence: 99%