A construction principle for tight and minimal triangulations of manifolds

Burton, Benjamin A., Datta, Basudeb, Singh, Nitin and Spreer, Jonathan (2016) A construction principle for tight and minimal triangulations of manifolds. Experimental Mathematics, . doi:10.1080/10586458.2016.1212747


Author Burton, Benjamin A.
Datta, Basudeb
Singh, Nitin
Spreer, Jonathan
Title A construction principle for tight and minimal triangulations of manifolds
Journal name Experimental Mathematics   Check publisher's open access policy
ISSN 1944-950X
1058-6458
Publication date 2016-10-04
Sub-type Article (original research)
DOI 10.1080/10586458.2016.1212747
Open Access Status Not yet assessed
Total pages 15
Place of publication New York, NY, United States
Publisher Taylor & Francis
Language eng
Formatted abstract
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 article, 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 2d−1d/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.
Keyword (Embeddings of) abstract simplicial complexes
Combinatorial manifold
Stacked sphere
Strongly minimal triangulation
Tight triangulation
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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