Minimum ideal triangulations of hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: Minimum ideal triangulations of hyperbolic 3-manifolds
zbMATH Open document ID: 0722.52008
DOI: 10.1007/BF02574680
author: William B. Sherman
author: Colin Adams
published in: Discrete \& Computational Geometry
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131149
review text: Let \(\sigma\) (n), resp. \(\sigma_{or}(n)\) denote the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volume n- cusped hyperbolic 3-manifold, resp. such an orientable one. The authors show that \(2n-1\leq \sigma (n)\leq \sigma_{or}(n)\leq 4n-4\) for \(n\geq 5\), and that \(\sigma_{or}(n)\geq 2n\) for all n. For \(n=1,2,3,4,5\) they give correct values of \(\sigma\) (n) and \(\sigma_{or}(n)\) and the corresponding manifolds in tables. \(\sigma (1)=1\), \(\sigma (2)=2\), \(\sigma (3)=4\), \(\sigma (4)=6\) have unique nonorientable realizations. \(\sigma (5)=\sigma_{or}(5)=10\) corresponds to one orientable manifold (tabulated) and at least two non-orientable ones. \(\sigma_{or}(1)=2\) and \(\sigma_{or}(2)=4\) each has two realizations. \(\sigma_{or}(3)=6\) and \(\sigma_{or}(4)=8\) appear with examples by link complements. The proof of minimality is simple combinatorial, except \(\sigma (5)=10\) which is not detailed here. These investigations are in connection with those of Thurston's circle. The reviewer lacks a reference to the paper of \textit{S. V. Matveev} and \textit{A. T. Fomenko} [Russ. Math. Surv. 43, No.1, 3-24 (1988); translation from Usp. Mat. Nauk. 43, No.1, 5-22 (1988; Zbl 0671.58008)].
reviewed by: Emil Molnár
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 4190576
zbMATH Keywords: n-cusped hyperbolic 3-manifold
MaRDI profile type: Publication
cites work: Thrice-Punctured Spheres in Hyperbolic 3-Manifolds
cites work: Q3769628
cites work: The Noncompact Hyperbolic 3-Manifold of Minimal Volume
cites work: Volumes of N -Cusped Hyperbolic 3-Manifolds
cites work: Hyperbolic Invariants of Knots and Links
cites work: Euclidean decompositions of noncompact hyperbolic manifolds
cites work: Q3828731
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: Hyperbolic structures on 3-manifolds. I: Deformation of acylindrical manifolds
cites work: Rotation Distance, Triangulations, and Hyperbolic Geometry
Recommended article: The orientable cusped hyperbolic \(3\)-manifolds of minimum volume
Recommended article: A census of cusped hyperbolic 3-manifolds
Recommended article: Small Hyperbolic 3-Manifolds With Geodesic Boundary
Recommended article: The Noncompact Hyperbolic 3-Manifold of Minimal Volume
Recommended article: Q4299371
Recommended article: Hyperbolic 4-manifolds, colourings and mutations
Recommended article: A linear bound on the tetrahedral number of manifolds of bounded volume (after Jørgensen and Thurston)
Recommended article: On minimal ideal triangulations of cusped hyperbolic 3‐manifolds
Recommended article: Q5396774
Recommended article: Hyperbolic four-manifolds with one cusp
description: scientific article; zbMATH DE number 4190576