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A Construction Principle for Tight and Minimal Triangulations of Manifolds

Burton, Benjamin A and Datta, Basudeb and Singh, Nitin and Spreer, Jonathan (2018) A Construction Principle for Tight and Minimal Triangulations of Manifolds. In: EXPERIMENTAL MATHEMATICS, 27 (1). pp. 22-36.

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Official URL: https://dx.doi.org/ 10.1080/10586458.2016.1212747

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 stronglyminimal 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 computerfriendly 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)(sic)d-2(sic)!(d-1)/2(sic)! 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.

Item Type: Journal Article
Additional Information: Copyright of this article belong to TAYLOR & FRANCIS INC, 530 WALNUT STREET, STE 850, PHILADELPHIA, PA 19106 USA
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 04 Jul 2018 14:42
Last Modified: 04 Jul 2018 14:42
URI: http://eprints.iisc.ac.in/id/eprint/60154

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