Stacked polytopes and tight triangulations of manifolds

Effenberger, Felix

doi:10.1016/j.jcta.2011.03.003

Cited by 20 publications

(36 citation statements)

References 52 publications

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“…We believe that theorem (b) is valid for dimension 2k + 1 as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.…”

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confidence: 79%

A tightness criterion for homology manifolds with or without boundary

Bagchi

2015

European Journal of Combinatorics

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A simplicial complex X is said to be tight with respect to a field F if X is connected and, for every induced subcomplex Y of X, the linear map H * (Y ; F) → H * (X; F) (induced by the inclusion map) is injective. This notion was introduced by Kühnel in [10]. In this paper we prove the following two combinatorial criteria for tightness. (a) Any (k + 1)neighbourly k-stacked F-homology manifold with boundary is F-tight. Also, (b) any F-orientable (k + 1)-neighbourly k-stacked F-homology manifold without boundary is F-tight, at least if its dimension is not equal to 2k + 1.The result (a) appears to be the first criterion to be found for tightness of (homology) manifolds with boundary. Since every (k + 1)neighbourly k-stacked manifold without boundary is, by definition, the boundary of a (k + 1)-neighbourly k-stacked manifold with boundary -and since we now know several examples (including two infinite families) of triangulations from the former class -theorem (a) provides us with many examples of tight triangulated manifolds with boundary.The second result (b) generalizes a similar result from [2] which was proved for a class of combinatorial manifolds without boundary. We believe that theorem (b) is valid for dimension 2k + 1 as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.

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“…We believe that theorem (b) is valid for dimension 2k + 1 as well. Except for this lacuna, this result answers a recent question of Effenberger [8] affirmatively.…”

supporting

confidence: 79%

A tightness criterion for homology manifolds with or without boundary

Bagchi

2015

European Journal of Combinatorics

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“…In [12], Lutz, Sulanke and Swartz conjectured that, for d ≥ 3, all tightneighbourly triangulated d-manifolds are tight. Using Novik-Swartz's result, Effenberger [9] proved this conjecture for d ≥ 4. Here, as a consequence of Theorem 1.3 and Proposition 2.5 below, we prove the Lutz-Sulanke-Swartz conjecture in the remaining case d = 3.…”

Section: Introduction and Resultsmentioning

confidence: 93%

“…Nonetheless, they have so far evaded complete combinatorial characterisation. From [9] and [3], we know the following: Bistellar flips or Pachner moves are ways of replacing a combinatorial triangulation of a piecewise linear manifold with another such triangulation of the same manifold. In dimension two, we have the following bistellar moves:…”

Section: Tight-neighbourly and Tight Triangulated Manifoldsmentioning

confidence: 99%

Separation index of graphs and stacked 2-spheres

Burton

Datta

Singh

et al. 2015

Journal of Combinatorial Theory, Series A

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In 1987, Kalai proved that stacked spheres of dimension d ≥ 3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d = 2. In this article, we give a characterisation of stacked 2-spheres using what we call the separation index. Namely, we show that the separation index of a triangulated 2-sphere is maximal if and only if it is stacked. In addition, we prove that, amongst all n-vertex triangulated 2-spheres, the separation index is minimised by some n-vertex flag sphere for n ≥ 6. Furthermore, we apply this characterisation of stacked 2-spheres to settle the outstanding 3-dimensional case of the Lutz-Sulanke-Swartz conjecture that "tight-neighbourly triangulated manifolds are tight". For dimension d ≥ 4, the conjecture has already been proved by Effenberger following a result of Novik and Swartz.MSC 2010 : 57Q15, 57M20, 05C40.

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“…Firstly, we have the following result. Theorem 2.3 (Effenberger [14]). For d = 3, the neighbourly members of K(d) are tight.…”

Section: Tight and Tight-neighbourly Triangulationsmentioning

confidence: 99%

A Construction Principle for Tight and Minimal Triangulations of Manifolds

Burton

Datta

Singh

et al. 2016

Experimental Mathematics

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Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions ≤ 3. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension.In this paper, we present a computer-friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated d-manifolds, with 2 d−1 ⌊d/2⌋!⌊(d−1)/2⌋! isomorphically distinct members for each dimension d ≥ 2. While we still do not know if there are infinitely many tight triangulations in a fixed dimension d > 2, this result shows that there are abundantly many.MSC 2010 : 57Q15, 57R05.

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Stacked polytopes and tight triangulations of manifolds

Cited by 20 publications

References 52 publications

A tightness criterion for homology manifolds with or without boundary

A tightness criterion for homology manifolds with or without boundary

Separation index of graphs and stacked 2-spheres

A Construction Principle for Tight and Minimal Triangulations of Manifolds

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