A Construction Principle for Tight and Minimal Triangulations of Manifolds

Journal of Combinatorial Theory, Series A

Singh

et al. 2015

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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.

Section: Novik and Swartzmentioning

confidence: 93%

Separation index of graphs and stacked 2-spheres

Burton

Journal of Combinatorial Theory, Series A

Singh

et al. 2015

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“…In [7], Z 2 -tight triangulations of (S 2 × − S 1 ) #k were constructed for k = 1, 30, 99, 208, 357 and 546. However, we do not know any F-tight triangulations of (S 2 × S 1 ) #k .…”

Section: Introductionmentioning

confidence: 99%

A characterization of tightly triangulated 3-manifolds

Bagchi

European Journal of Combinatorics

Spreer

2017

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For a field F, the notion of F-tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that any F-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only F-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is F-tight if and only if it is F-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is F-tight if and only if it is F-orientable, neighbourly and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension ≤ 3.MSC 2010 : 57Q15, 57R05.

“…In Section 6, we present some examples to show that converses and generalizations of several results proved here are not true. Recently, it has been found in [8] that (20m + 9)-vertex tight, triangulated 3-manifolds exist for all m ≤ 5. The paper [8] lists 76 non-isomorphic tight triangulations with these parameters including Walkup's 9-vertex 3-dimensional Klein bottle.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, it has been found in [8] that (20m + 9)-vertex tight, triangulated 3-manifolds exist for all m ≤ 5. The paper [8] lists 76 non-isomorphic tight triangulations with these parameters including Walkup's 9-vertex 3-dimensional Klein bottle. Existence of these examples makes it natural to pose the following conjecture.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tight triangulations of closed 3-manifolds

Bagchi

European Journal of Combinatorics

Spreer

2016

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A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic.Next let F be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is F-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an F-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an F-tight triangulation of a closed 3-manifold has n vertices and first Betti number β 1 , then (n − 4)(617n − 3861) ≤ 15444β 1 . Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.MSC 2010 : 57Q15, 57R05.