Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Bachman, David

doi:10.1007/s00208-012-0802-4

Cited by 30 publications

(31 citation statements)

References 28 publications

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“…Bachman [1] has recently announced similar examples using different techniques; where we use bicompressible surfaces to compare two Heegaard splittings, he uses incompressible surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2. Given a genus k 2 Heegaard splitting (Σ, H − , H + ) for a closed 3-manifold M , the flip genus of Σ is greater than or equal to min{2k, 1 2 d(Σ)}.…”

Section: Introductionmentioning

confidence: 99%

“…Let Σ be a genus k 2 Heegaard splitting of a 3-manifold with a single boundary component and let Σ be the result of boundary stabilizing Σ. Then the stable genus of Σ and Σ is greater than or equal to min{2k, 1 2 d(Σ)}.…”

Section: Introductionmentioning

confidence: 99%

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Bounding the stable genera of Heegaard splittings from below

Johnson

2010

Journal of Topology

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We describe for each positive integer k a 3‐manifold with Heegaard surfaces of genus 2k and 2k−1 such that any common stabilization of these two surfaces has genus at least 3k−1. We also find, for every k, a 3‐manifold with boundary admitting Heegaard splittings of genus k and k+1 whose stable genus is 2k, and for every positive n, a 3‐manifold that has n pairwise nonisotopic Heegaard splittings of the same genus all of which are stabilized.

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“…Bachman [1] has recently announced similar examples using different techniques; where we use bicompressible surfaces to compare two Heegaard splittings, he uses incompressible surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2. Given a genus k 2 Heegaard splitting (Σ, H − , H + ) for a closed 3-manifold M , the flip genus of Σ is greater than or equal to min{2k, 1 2 d(Σ)}.…”

Section: Introductionmentioning

confidence: 99%

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Bounding the stable genera of Heegaard splittings from below

Johnson

2010

Journal of Topology

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“…A long standing question in Heegaard splittings asks what is the minimal genus of † 00 in terms of the genera of † and † 0 . Examples of Heegaard splittings that required many stabilizations were presented by Bachman [1], Hass, Thompson and Thurston [3] and Johnson [5].…”

Section: Introductionmentioning

confidence: 99%

Flipping bridge surfaces and bounds on the stable bridge number

Johnson

Tomova

2011

Algebr. Geom. Topol.

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We show that if K is a knot in S 3 and † is a bridge sphere for K with high distance and 2n punctures, the number of perturbations of K required to interchange the two balls bounded by † via an isotopy is n. We also construct a knot with two different bridge spheres with 2n and 2n 1 bridges respectively for which any common perturbation has at least 3n 4 bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.57M25, 57M27, 57M50

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“…Just like the incompressible and strongly irreducible surfaces before them, critical surfaces and topologically minimal surfaces in general have been used to prove long-standing conjectures that had remained unresolved for many years. For example, Bachman used these surfaces to prove The Gordon Conjecture 1 and to provide counterexamples to the Stabilization Conjecture 2 [3].…”

Section: Introductionmentioning

confidence: 99%

An infinite family of links with critical bridge spheres

Rodman

2018

Algebr. Geom. Topol.

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ACKNOWLEDGEMENTSThis thesis represents the work, support, encouragement, and love of countless family, friends, mentors, teachers, and pastors, each responsible for a piece of my development as a learner, a teacher, and a human being. Though it is tempting to try, it would be impossible to recognize every one of them here, so I will have to limit these acknowledgments to those individuals who have specifically shaped me as a mathematician. Soli Deo gloriaii ABSTRACT A closed, orientable, splitting surface in an oriented 3-manifold is a topologically minimal surface of index n if its associated disk complex is (n−2)-connected but not (n − 1)-connected. A critical surface is a topologically minimal surface of index 2. In this thesis, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.iii PUBLIC ABSTRACTA knot can be thought of as the result of taking a shoe string, twisting it up in some way, and gluing the ends together. A link is a set of any number of knots which may be twisted and wound up around each other. When a sphere intersects a link in such a way that the link is cut into pieces such that each piece has exactly one maximum or minimum point, the sphere is said to be a bridge sphere for the link.Making extensive use of a tool called a compressing disk, we can study how a link and its bridge sphere intersect each other by associating to them a multidimensional space Γ called a simplicial complex. A natural question is, does Γ have any 2-dimensional holes in it? In other words, is every circle in Γ the boundary of a disk? If Γ is connected, but it contains a 2-dimensional hole, then the bridge sphere is called critical.We use an equivalent and more tangible definition of a critical bridge sphere defined by how compressing disks on one side of the sphere intersect compressing disks on the other side. In this thesis, we study an infinite family of links which can be laid out in a specific plat pattern. We examine their bridge spheres and how their compressing disks intersect each other to conclude that each of these bridge spheres is critical.

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Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Cited by 30 publications

References 28 publications

Bounding the stable genera of Heegaard splittings from below

Bounding the stable genera of Heegaard splittings from below

Flipping bridge surfaces and bounds on the stable bridge number

An infinite family of links with critical bridge spheres

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