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229
The asymptotic behaviour of Heegaard genus
, 2004
"... Heegaard splittings have recently been shown to be related to a number of important conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture [3]. Of particular importance is the rate at which the Heegaard genus of finite ..."
Abstract
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Cited by 11 (1 self)
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Heegaard splittings have recently been shown to be related to a number of important conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture [3]. Of particular importance is the rate at which the Heegaard genus
ON THE HEEGAARD GENUS OF CONTACT 3-MANIFOLDS
"... ABSTRACT. It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus [8]) of 3-manif ..."
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ABSTRACT. It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus [8]) of 3
Degeneration of Heegaard genus, a survey
- In Workshop on Heegaard Splittings
, 2007
"... Abstract. We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface × I, and (3) ..."
Abstract
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Cited by 1 (0 self)
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Abstract. We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface × I, and (3
THE HEEGAARD GENUS OF AMALGAMATED 3-MANIFOLDS
, 2003
"... When studying Haken 3-manifolds, one is led naturally to the following construction: the amalgamation of two 3-manifolds M and M ′ via a homeomorphism between their boundaries. In this paper, we study the behaviour of Heegaard genus under this operation. We show that, provided the gluing homeomorphi ..."
Abstract
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Cited by 19 (0 self)
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When studying Haken 3-manifolds, one is led naturally to the following construction: the amalgamation of two 3-manifolds M and M ′ via a homeomorphism between their boundaries. In this paper, we study the behaviour of Heegaard genus under this operation. We show that, provided the gluing
The Heegaard genus of bundles over S¹
, 2007
"... This paper explores connections between Heegaard genus, minimal surfaces, and pseudo-Anosov monodromies. Fixing a pseudo-Anosov map φ and an integer n, let Mn be the 3–manifold fibered over S 1 with monodromy φ n. JH Rubinstein showed that for a large enough n every minimal surface of genus at most ..."
Abstract
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Cited by 1 (0 self)
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This paper explores connections between Heegaard genus, minimal surfaces, and pseudo-Anosov monodromies. Fixing a pseudo-Anosov map φ and an integer n, let Mn be the 3–manifold fibered over S 1 with monodromy φ n. JH Rubinstein showed that for a large enough n every minimal surface of genus at most
Alternate Heegaard genus bounds distance
- GEOM. AND TOPOLOGY
, 2006
"... Suppose M is a compact orientable irreducible 3–manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundarystabilized copy of P or the distance d.P / 2genus.Q/. More generally, if P and Q are bicompressible but weakly incompressible connected clo ..."
Abstract
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Cited by 48 (3 self)
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Suppose M is a compact orientable irreducible 3–manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundarystabilized copy of P or the distance d.P / 2genus.Q/. More generally, if P and Q are bicompressible but weakly incompressible connected
Heegaard genus formula for Haken manifolds
"... Suppose M is a compact orientable 3-manifold and Q ⊂ M a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M − Q with respect to the path metric by M 1,..., M k. Denote the smallest possible genus of a Heegaard splitting of M, or M j ..."
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Cited by 7 (1 self)
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Suppose M is a compact orientable 3-manifold and Q ⊂ M a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M − Q with respect to the path metric by M 1,..., M k. Denote the smallest possible genus of a Heegaard splitting of M, or M j
Quantum invariants provide sharp Heegaard genus bounds
, 801
"... Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group. 1 ..."
Abstract
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Cited by 1 (1 self)
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Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group. 1
Results 1 - 10
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229
