Non-isotopic Heegaard splittings of Seifert fibered spaces

Bachman, David; Derby-Talbot, Ryan

doi:10.2140/agt.2006.6.351

Cited by 9 publications

(20 citation statements)

References 13 publications

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“…For toroidal manifolds, Morimoto and Sakuma [26] have found a number of tunnel number one knot complements and Bachman and Derby-Talbot [3] have found Seifert fibered spaces with an infinite number of homeomorphic, non-isotopic Heegaard splittings. For all these Heegaard splittings, the image of i has infinite index.…”

Section: The Mapping Class Groupmentioning

confidence: 99%

Mapping Class Groups of Heegaard Splittings

Johnson

Rubinstein

2013

J. Knot Theory Ramifications

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Section: The Mapping Class Groupmentioning

confidence: 99%

Mapping Class Groups of Heegaard Splittings

Johnson

Rubinstein

2013

J. Knot Theory Ramifications

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“…To complete the proof, isotope F so as to remove any loops that are trivial in both F and S. Bachman and Derby-Talbot [2006] pointed out that after these trivial loops are removed, S \ F is a pair of compressing disks for F whose boundaries, when made transverse, intersect at four points. These compressing disks are on opposite sides of F and are contained in the regular neighborhood R. Any compressing disk for F \ R is disjoint from each of the disks in S \ F. Because ( , H 1 , H 2 ) is strongly irreducible and F is isotopic to , the surface \ R must be incompressible in M \ R.…”

Section: Toroidal Summandsmentioning

confidence: 99%

“…Bachman and Derby-Talbot [2006] showed that any Seifert fibered space that admits a strongly irreducible horizontal splitting admits infinitely many isotopy classes of horizontal splittings. We improve their analysis to show the following:…”

Section: Introductionmentioning

confidence: 99%

“…Weidmann has shown, in the appendix of [Bachman and Derby-Talbot 2006], that every circle bundle contains a unique irreducible Heegaard splitting (up to isotopy). The only circle bundles with strongly irreducible Heegaard splittings are circle bundles over the circle (all of which are lens spaces) and the circle bundle over the torus with Euler number one.…”

Section: Introductionmentioning

confidence: 99%

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Horizontal Heegaard splittings of Seifert fibered spaces

Johnson¹

2010

Pacific J. Math.

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We show that if an orientable Seifert fibered space M with an orientable genus g base space admits a strongly irreducible horizontal Heegaard splitting, then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of ‫ޚ‬ 2g . This correspondence is determined by the slopes of intersection of each Heegaard splitting with a set of 2g incompressible tori in M. We also show there are Seifert fibered spaces with infinitely many nonisotopic Heegaard splittings that determine Nielsen equivalent generating systems for the fundamental group of M.

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“…In Section 5 we survey known work that gives quite satisfactory restrictions to phenomena (1) and (2). We also give an extended example: Dehn filling on a torus knot exterior, for which we have almost complete knowledge.…”

Section: Introductionmentioning

confidence: 99%

The Heegaard structure of Dehn filled manifolds

Moriah¹,

Sedgwick²

2007

Geometry and Topology Monographs

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233arXiv version: fonts, pagination and layout may vary from GTM published versionThe Heegaard structure of Dehn filled manifolds YOAV MORIAH ERIC SEDGWICKWe expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena occur when filling most torus knot exteriors. This latter thread yields sufficient (and perhaps necessary) conditions for the occurrence of phenomena (3) and (4). 57N10; 57M27 IntroductionLet X be a knot manifold, that is a compact, orientable and irreducible 3-manifold with a single torus boundary component. There are many results demonstrating that most of the manifolds obtained by filling inherit properties from the knot manifold. We would also expect the Heegaard structure of filled manifolds to be closely related to the Heegaard structure of the knot manifold. For example, it is easy to see that every Heegaard surface for the knot manifold is a Heegaard surface for each filled manifold. In particular, this implies that the Heegaard genus of X is an upper bound on the genus of each filled manifold. However, the Heegaard structure of a filled manifold can differ from that of the knot manifold. Here are four ways that this could occur:(1) Heegaard genus decreases. By a new Heegaard surface, we mean that a filled manifold contains a Heegaard surface that is not isotopic (in the filled manifold) to a Heegaard surface for the knot manifold X . When the genus decreases (1), the filled manifold has a Heegaard surface of lower genus than every Heegaard surface for X . Indeed, it is a new Heegaard surface. So, restricting (2) also restricts (1).In each of these cases, we would like to either demonstrate that the set of fillings for which the phenomenon occurs is special, for example finite, a line of slopes, and/or conclude that the Heegaard surface(s) in question are special in some regard, for example γ -primitive, padded, or boundary stabilized.In Section 5 we survey known work that gives quite satisfactory restrictions to phenomena (1) and (2). We also give an extended example: Dehn filling on a torus knot exterior, for which we have almost complete knowledge. We are able to completely specify the fillings for which each of these four phenomena occur. This also illustrates sufficient conditions for (3) and (4) to occur. AcknowledgementsWe would like to thank Ryan Derby-Talbot, David Bachman, Jesse Johnson and Jennifer Schultens for helpful discussions. Also thanks to DePaul CTI and the Mathematics Department of the Technion for their hospitality. Background 2.1 Dehn fillin...

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Non-isotopic Heegaard splittings of Seifert fibered spaces

Cited by 9 publications

References 13 publications

Mapping Class Groups of Heegaard Splittings

Mapping Class Groups of Heegaard Splittings

Horizontal Heegaard splittings of Seifert fibered spaces

The Heegaard structure of Dehn filled manifolds

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