Combinatorics of triangulations of 3-manifolds

Luo, Feng; Stong, Richard

doi:10.1090/s0002-9947-1993-1134759-6

Cited by 14 publications

(11 citation statements)

References 3 publications

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“…Of course, we know that the Euler number can give no such information in three dimensions, but Luo and Stong [27] have given one result in this direction. They show that any triangulation with z < 8 (n < 9/2) must be a triangulation of (some quotient of) S 3 .…”

Section: Bounds On the Combinatorics Of Triangulationsmentioning

confidence: 99%

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The Geometry of Bubbles and Foams

Sullivan

1999

Foams and Emulsions

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Abstract. We consider mathematical models of bubbles, foams and froths, as collections of surfaces which minimize area under volume constraints. The resulting surfaces have constant mean curvature and an invariant notion of equilibrium forces. The possible singularities are described by Plateau's rules; this means that combinatorially a foam is dual to some triangulation of space. We examine certain restrictions on the combinatorics of triangulations and some useful ways to construct triangulations. Finally, we examine particular structures, like the family of tetrahedrally close-packed structures. These include the one used by Weaire and Phelan in their counterexample to the Kelvin conjecture, and they all seem useful for generating good equal-volume foams.

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Section: Bounds On the Combinatorics Of Triangulationsmentioning

confidence: 99%

“…We can apply these iteratively (as shown in [27]) to get a sequence of triangulations with arbitrary z > 8 in the limit, on any three-manifold.…”

Section: Bounds On the Combinatorics Of Triangulationsmentioning

confidence: 99%

The Geometry of Bubbles and Foams

Sullivan

1999

Foams and Emulsions

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“…This is equal to the average of the orders of edges of K, where the order of an edge is the number of faces incident to that edge. Feng Luo and Richard Stong showed in [1] that for a closed 3-manifold M , the average edge order being small implies that the topology of M is fairly simple and restricts the triangulation K of M . In fact, they proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%

The average edge order of triangulations of 3-manifolds with boundary

Tamura¹

1998

Trans. Amer. Math. Soc.

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“…It implies that the geometric realization of ∆(G) is a triangulation of 3-dimensional sphere S 3 with face vector (12, 45, 66, 33) (see [11]), so G is a Gorenstein graph. The proof of the theorem is complete.…”

Section: Case 3: α(G)mentioning

confidence: 99%

Buchsbaumness of the second powers of edge ideals

Hoang

Trung

2018

J. Algebra Appl.

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Abstract. We graph-theoretically characterize the class of graphs G such that I(G) 2 are Buchsbaum. IntroductionThroughout this paper let G = (V (G), E(G)) be a finite simple graph without isolated vertices. An independent set in G is a set of vertices no two of which are adjacent to each other. The size of the largest independent set, denoted by α(G), is called the independence number of G. A graph is called well-covered if every maximal independent set has the same size. A well-covered graph G is a member of the class W 2 if the remove any vertex of G leaves a well-covered graph with the same independence number as G (see e.g. [14]).Let R = K[x 1 , . . . , x n ] be a polynomial ring of n variables over a given field K. Let G be a simple graph on the vertex set V (G) = {x 1 , . . . , x n }. We associate to the graph G a quadratic squarefree monomial idealwhich is called the edge ideal of G. We say that G is Cohen-Macaulay (resp. Gorenstein) if I(G) is a Cohen-Macaulay (resp. Gorenstein) ideal. It is known that G is well-covered whenever it is Cohen-Macaulay (see e.g. [20, Proposition 6.1.21]) and G is in W 2 whenever it is Gorenstein (see e.g. [10, Lemma 2.5]). It is a wide open problem to characterize graph-theoretically the Cohen-Macaulay (resp. Gorenstein) graphs. This problem was considered for certain classes of graphs (see [5,6,9,10]). Generally, we cannot read off the Cohen-Macaulay and Gorenstein properties of G just from its structure because these properties in fact depend on the characteristic of the base field K (see [20, Exercise 5.3.31] and [10, Proposition 2.1]).If we move on to the higher powers of I(G), then we can graph-theoretically characterize G such that I(G) m is Cohen-Macaulay (or Buchsbaum, or generalized CohenMacaulay) for some m 3 (and for all m 1) (see [4,15,19]

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Combinatorics of triangulations of 3-manifolds

Abstract: Abstract. In this paper, we study the average edge order of triangulations of closed 3-manifolds and show in particular that the average edge order being less than 4.5 implies that triangulation is on the 3-sphere.

Cited by 14 publications

References 3 publications

The Geometry of Bubbles and Foams

The Geometry of Bubbles and Foams

The average edge order of triangulations of 3-manifolds with boundary

Buchsbaumness of the second powers of edge ideals

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