Heegaard Genus of the Connected Sum of M-small Knots

Kobayashi, Teiichi; Rieck, Yo'av

doi:10.4310/cag.2006.v14.n5.a8

Cited by 22 publications

(24 citation statements)

References 24 publications

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“…Therefore, for some i, S is isotopic to F i . In Rieck and Kobayashi [13,Proposition 2.13] it was shown that if Σ untelescopes to the essential surface ∪ i F i , then Σ weakly reduces to any connected separating component of ∪ i F i ; therefore Σ weakly reduces to S. This proves the first part of Claim 1.…”

Section: Claimmentioning

confidence: 73%

The Heegaard genus of bundles over S¹

Brittenham¹

2007

Geometry and Topology Monographs

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The Heegaard genus of bundles over S 1 MARK BRITTENHAM YO'AV RIECK This paper explores connections between Heegaard genus, minimal surfaces, and pseudo-Anosov monodromies. Fixing a pseudo-Anosov map φ and an integer n, let M n 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 h in M n is homotopic into a fiber; as a consequence Rubinstein concludes that every Heegaard surface of genus at most h for M n is standard, that is, obtained by tubing together two fibers. We prove this result and also discuss related results of Lackenby and Souto.

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Section: Claimmentioning

confidence: 73%

The Heegaard genus of bundles over S¹

Brittenham¹

2007

Geometry and Topology Monographs

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“…It is well known that E(4 1 ) admits a genus 2 Heegaard splitting. By amalgamating a genus 2 Heegaard splitting for E(4 1 ) with a genus 2 Heegaard splitting of E(3 1 ) ∪ E(L) we obtain a weakly reducible Heegaard splitting of M; by [9] (see also [6,Lemma 2.7] for a more general statement) this Heegaard splitting has genus 3. This establishes the existence of weakly reducible minimal genus Heegaard splittings of M.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

Kobayashi

Rieck²

2009

Communications in Analysis and Geometry

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We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.The pioneering work of Casson and Gordon [1] shows that a minimal genus Heegaard splitting of an irreducible, non-Haken 3-manifold is necessarily strongly irreducible; by contrast, Haken [2] showed that a minimal genus (indeed, any) Heegaard splitting of a composite 3manifold is necessarily reducible, and hence weakly reducible. The following question of Moriah [8] is therefore quite natural: Question 1 ([8], Question 1.2). Can a 3-manifold M have both weakly reducible and strongly irreducible minimal genus Heegaard splittings?We answer this question affirmatively:Theorem 2. There exist infinitely many closed, orientable 3-manifolds of Heegaard genus 3, each admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings.

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“…This bound is given in terms of the bridge number of the branch set with respect to the Heegaard surface for N given in Theorem 1.2, i.e., the projection of the invariant Heegaard surface for M . The definition of bridge number with respect to a Heegaard surface is given in Definition 12.1 (for a detailed discussion see, for example, [12] or [10]). We prove: For proving Theorem 1.1 we study the intersection of strongly irreducible Heegaard surfaces, that is, the intersection of Σ and its image under the involution f (Σ).…”

Section: Theorem 12 (Genus Of Double Covers) Let M Be An Irreduciblmentioning

confidence: 99%

Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers

Rieck¹,

Rubinstein

2009

Communications in Analysis and Geometry

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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 Heegaard genus of N :We also get a bound on the complexity of the branch set in terms of g(Σ). If we assume that M is non-Haken, by Casson and Gordon [3] we may replace g(Σ) by g(M ) in all the statements above.

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Heegaard Genus of the Connected Sum of M-small Knots

Cited by 22 publications

References 24 publications

The Heegaard genus of bundles over S¹

The Heegaard genus of bundles over S¹

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

Invariant Heegaard surfaces in manifolds with involutions and the Heegaard genus of double covers

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