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Results 1 - 10
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171
ALMOST NORMAL HEEGAARD SURFACES
, 2003
"... Abstract. We present a new and shorter proof of Stocking’s result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3– manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein on normal spheres. Both proofs are based on the “ ..."
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Cited by 2 (0 self)
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Abstract. We present a new and shorter proof of Stocking’s result that any strongly irreducible Heegaard surface of a closed orientable triangulated 3– manifold is isotopic to an almost normal surface. We also re-prove a result of Jaco and Rubinstein on normal spheres. Both proofs are based
CRITICAL HEEGAARD SURFACES
, 2002
"... Abstract. In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, t ..."
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Cited by 17 (8 self)
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Abstract. In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical
HEEGAARD SURFACES AND THE DISTANCE OF AMALGAMATION
, 2008
"... Let M1 and M2 be orientable irreducible 3–manifolds with connected boundary and suppose ∂M1 ∼ = ∂M2. Let M be a closed 3–manifold obtained by gluing M1 to M2 along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then M is not homeomorphic to S 3 and all small-ge ..."
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Cited by 6 (3 self)
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-genus Heegaard splittings of M are standard in a certain sense. In particular, g(M) = g(M1) + g(M2) − g(∂Mi), where g(M) denotes the Heegaard genus of M. This theorem can also be extended to manifolds with multiple boundary components.
Lifted Heegaard Surfaces and Virtually Haken Manifolds
"... In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [ ..."
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In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements
Thin position with respect to a Heegaard surface
- ALMOST NORMAL SURFACES IN KNOT COMPLEMENTS 23
, 2001
"... We present a definition of thin position for a knot in a 3-manifold with respect to a Heegaard surface, motivated by Scharlamenn and Thompson's definition of thin position for 3-manifolds [ST94], and Gabai's definition of thin position for knots in S3 [Gab87]. We then show that if a knot ..."
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Cited by 4 (1 self)
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We present a definition of thin position for a knot in a 3-manifold with respect to a Heegaard surface, motivated by Scharlamenn and Thompson's definition of thin position for 3-manifolds [ST94], and Gabai's definition of thin position for knots in S3 [Gab87]. We then show that if a
K-STABLE EQUIVALENCE FOR KNOTS IN HEEGAARD SURFACES
, 2009
"... Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S′ , K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope. ..."
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Cited by 1 (0 self)
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Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S′ , K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.
Invariant heegaard surfaces in manifolds with involutions and the heegaard genus of double covers
"... ABSTRACT. Let M be a 3-manifold admitting a strongly irreducible Heegaard surface Σ and f: M → M an involution. We construct an invariant Heegaard surface for M of genus at most 8g(Σ) − 7. As a consequence, given a (possibly branched) double cover π: M → N we obtain the following bound on the Heega ..."
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Cited by 3 (1 self)
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ABSTRACT. Let M be a 3-manifold admitting a strongly irreducible Heegaard surface Σ and f: M → M an involution. We construct an invariant Heegaard surface for M of genus at most 8g(Σ) − 7. As a consequence, given a (possibly branched) double cover π: M → N we obtain the following bound
CRITICAL HEEGAARD SURFACES AND INDEX 2 MINIMAL SURFACES
, 2002
"... Abstract. This paper contains the motivation for the study of critical surfaces in [2]. In that paper, the only justification given for the definition of this new class of surfaces is the strength of the results. However, when viewed as the topological analogue to index 2 minimal surfaces, critical ..."
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Cited by 1 (1 self)
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surfaces become quite natural. 1. Introduction. It is a standard exercise in 3-manifold topology to show that every manifold admits Heegaard splittings of arbitrarily high genus. Hence, a “random ” Heegaard splitting does not say much about the topology of the manifold in which it sits. To use Heegaard
Results 1 - 10
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171
