An infinite family of tight triangulations of manifolds

Datta, Basudeb; Singh, Nitin

doi:10.1016/j.jcta.2013.08.005

Cited by 11 publications

(17 citation statements)

References 19 publications

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“…(b) For each d ≥ 3, Datta and Singh [7] constructed two non-isomorphic triangulated d-manifolds with boundary, named M d n and N d n . Both are 1-stacked 2-neighbourly, with d 2 +3d+1 vertices.…”

Section: Sketch Of Proofmentioning

confidence: 99%

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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Section: Sketch Of Proofmentioning

confidence: 99%

A tightness criterion for homology manifolds with or without boundary

Bagchi

2015

European Journal of Combinatorics

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“…. , (f) in the proof of Lemma 4.4 in [13] it was shown that condition (i) is satisfied in this case.…”

Section: D−1 · ⌊D/2⌋! · ⌊(D − 1)/2⌋! Non-isomorphic Tight D-manifoldsmentioning

confidence: 72%

“…In this section we describe a representation of a weak pseudomanifold K in terms of its dual graph G and its (stacked) vertex links, given by a collection of trees T . This representation was first used by the second and third authors [13]. The complex K(G, T ), defined below, is the central object of our construction principle for tight manifolds presented in this article.…”

Section: The Complex K(g T )mentioning

confidence: 99%

“…We denote the facet {i : u ∈ T i } byû for u ∈ V (G). Our construction is based on the following result from [13]. Then K(G, T ) is a neighbourly member of K(d + 1), with dual graph Γ(K(G, T )) isomorphic to G.…”

Section: The Complex K(g T )mentioning

confidence: 99%

“…Let X d n = ∂Y 1 . Since both n and d are odd, X d n is non-orientable by [13,Lemma 5.1]. (In fact, X d n triangulates the non-orientable S d−1 -bundle S d−1 × − S 1 over S 1 .)…”

Section: D−1 · ⌊D/2⌋! · ⌊(D − 1)/2⌋! Non-isomorphic Tight D-manifoldsmentioning

confidence: 99%

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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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Strongly minimal triangulations of (S 3×S 1)#3 and (S 3 S 1)#3

Singh

2015

Proc Math Sci

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Noncovalent interactions that are selective, directional, and strongly attractive can induce the self-assembly of predictable su-pramolecular aggregates. Tectons (from Greek, tekton, builder) of selected local symmetry, length and flexibility are used as simple assembly blocs of Lego type for construction of specialized networks. Several diamantoid networks were crystallized and studied by X-ray diffraction technique in order to verify the self-assembling properties of those novel tectons. These structures have in common the fonnation of large chambers whose size and shape follow directly from that of the tecton subunits. The network's large cavities so obtained use self-filling by mutual concatenation of similar networks to minimized the empty space in these structures. Besides the large chan1bers, large channels remain occupied by solvent molecules. Novel types of networks, derived from the dian1antoid type, were also characterized. Several new two-and three-dimensional networks were crystallized by using tectonic subunits of locally different primary symmetry. By modification of the flexibility and/or the tenninal group of the arms, new network types were promoted. This communication will present an overview of the subject as well as our most recent works in this area. Several years ago a series of dihydroxitetraalkyldisiloxanes has been studied by physical methods in order to find the structure-property relations for these compounds, which fom1 termothropic liquid crystalline phases. It was found that liquid crystalline properties are due to the formation of one-dimensional columnar hydrogen bonded associates in crystal and mesophase. These associates have pseudohexagonal packing in crystal and fonn mesophase with hexagonal type of columnar packing. In order to find other compounds inclined to form mesophases of the same type we performed conformational calculations and molecular modeling of molecular associates of dihydroxycarbosilanes and dihydroxydisiltianes. It was shown that conformational properties and associate formation for dihydroxycarbosilanes and dihydroxidisiloxanes is very close, but they are not the same for dihydroxydisiltianes. To support the close supramolecular structure of siloxane and carbosilane derivatives crystallization and X-ray analysis of dihydroxitetra-methyldicarbosalan was performed. The one-dimensional columnar associate in this crystal was found to be nearly the same as for disiloxane. DSC and multitemperature powder X-ray data are discussed together with the data on single crystal structure. PS06.05.10 X-RAY AND NEUTRON DIFFRACTION STUDY OF A CROWN ETHER-A CAUTIONARY TALE. R.H. Fenn, University of Portsmouth, S.A. Mason, ILL Grenoble, R.A. Palmer and B. Potter, Birkbeck College, London, O.S. Mills, P.M. Robinson and C. I. F. Watt, University of Manchester, UK Beware all Ye who regard a diffractometer as a black box and do not appreciate the beauty of crystallography. The compound C24H3o06.H20 is a water-bound adduct of the tetragol crown ether 9,10-dihydro,-10,10-dimethyl,-9...

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An infinite family of tight triangulations of manifolds

Cited by 11 publications

References 19 publications

A tightness criterion for homology manifolds with or without boundary

A tightness criterion for homology manifolds with or without boundary

A Construction Principle for Tight and Minimal Triangulations of Manifolds

Strongly minimal triangulations of (S 3×S 1)#3 and (S 3 S 1)#3

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