# A Construction Principle for Tight and Minimal Triangulations of Manifolds

@article{Burton2018ACP,
title={A Construction Principle for Tight and Minimal Triangulations of Manifolds},
author={Benjamin A. Burton and Basudeb Datta and Nitin Singh and Jonathan Spreer},
journal={Experimental Mathematics},
year={2018},
volume={27},
pages={22 - 36}
}
• Published 14 November 2015
• Mathematics
• Experimental Mathematics
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…
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## References

SHOWING 1-10 OF 37 REFERENCES
A CENSUS OF TIGHT TRIANGULATIONS
• Mathematics
• 2000
A triangulation of a manifold (or pseudomanifold) is called a tight triangulation if any simplexwise linear embedding into any Euclidean space is tight. Tightness of an embedding means that the
Tight triangulations of closed 3-manifolds
• Mathematics
Eur. J. Comb.
• 2016
An infinite family of tight triangulations of manifolds
• Mathematics
J. Comb. Theory, Ser. A
• 2013
Efficient Algorithms to Decide Tightness
• Mathematics, Computer Science
SoCG
• 2016
This article presents a new polynomial time procedure to decide tightness for triangulations of 3-manifolds – a problem which previously was thought to be hard.
The lower bound conjecture for 3- and 4-manifolds
For any closed connected d-manifold M let ](M) denote the set of vectors / (K)= (]0(K) ..... fd(K)), where K ranges over all triangulations of M and ]k(K) denotes the number of k-simplices of K. The
On Walkup’s class K(d) and a minimal triangulation of (S ⋉S)
• Mathematics
• 2013
For d ≥ 2, Walkup’s class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d − 1)-spheres. Kalai showed that for d ≥ 4, all connected members of K(d) are
Rigidity and the lower bound theorem 1
SummaryFor an arbitrary triangulated (d-1)-manifold without boundaryC withf0 vertices andf1 edges, define \gamma (C) = f_1 - df_0 + \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} }