Unstabilized dual Heegaard splittings of 3-manifolds

2016 ◽  
Vol 25 (06) ◽  
pp. 1650032 ◽  
Author(s):  
Kun Du
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Planar Surface

Let [Formula: see text] be a [Formula: see text]-manifold, [Formula: see text] be an essential planar surface which cuts [Formula: see text] into two 3-manifolds [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] [Formula: see text] is a Heegaard splitting of [Formula: see text], [Formula: see text] is the dual Heegaard splitting of [Formula: see text] and [Formula: see text] along [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, we give a condition of unstabilized dual Heegaard splittings of [Formula: see text]-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of [Formula: see text]-manifolds for any Hempel’s distance.

Local uniqueness of certain geodesics related to Heegaard splittings

2018 ◽  
Vol 27 (09) ◽  
pp. 1842003
Author(s):  
Liang Liang ◽  
Fengling Li ◽  
Fengchun Lei ◽  
Jie Wu
Keyword(s):  
Heegaard Splitting ◽  
Sufficient Condition ◽  
Heegaard Splittings ◽  
Local Uniqueness ◽  
Uniqueness Property

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

RECTANGLE CONDITION FOR COMPRESSION BODIES AND 2-FOLD BRANCHED COVERINGS

2012 ◽  
Vol 21 (08) ◽  
pp. 1250078
Author(s):  
JUNGSOO KIM ◽  
JUNG HOON LEE
Keyword(s):  
Heegaard Splitting ◽  
Level Surface ◽  
Heegaard Splittings ◽  
Branched Coverings ◽  
Link Complement ◽  
Strong Irreducibility

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S3 branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

3-Manifold invariants derived from the intersecting kernels

2016 ◽  
Vol 27 (13) ◽  
pp. 1650109
Author(s):  
Fengling Li ◽  
Fengchun Lei ◽  
Jie Wu
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Quotient Groups

The intersecting kernel of a Heegaard splitting [Formula: see text] for a compact orientable 3-manifold [Formula: see text] is the subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is the homomorphism induced by the inclusion [Formula: see text], [Formula: see text]. In the paper, we obtain some invariants of 3-manifolds [Formula: see text] from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

scholarly journals A distance for circular Heegaard splittings

2019 ◽  
Vol 28 (09) ◽  
pp. 1950059
Author(s):  
Kevin Lamb ◽  
Patrick Weed
Keyword(s):  
Critical Points ◽  
Morse Function ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Seifert Surface ◽  
Singular Foliation ◽  
Incompressible Surfaces ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Index 2

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.

scholarly journals MAPPING CLASS GROUPS OF HEEGAARD SPLITTINGS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350018 ◽  
Author(s):  
JESSE JOHNSON ◽  
HYAM RUBINSTEIN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Heegaard Splitting ◽  
Mapping Class ◽  
Connected Components ◽  
Heegaard Splittings ◽  
Ambient Manifold ◽  
Periodic Element ◽  
Heegaard Surface ◽  
Structural Theorems

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. We find examples of elements of the mapping class group that are periodic, reducible and pseudo-Anosov on the Heegaard surface, but are isotopy trivial in the ambient manifold. We prove structural theorems about the first two classes, in particular showing that if a periodic element is trivial in the mapping class group of the ambient manifold, then the manifold is not hyperbolic.

scholarly journals Parity criterion for unstabilized Heegaard splittings

2010 ◽  
Vol 149 (1) ◽  
pp. 115-125
Author(s):  
JUNG HOON LEE
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Branched Coverings ◽  
Dehn Twists ◽  
Heegaard Diagram ◽  
Parity Condition

AbstractWe give a parity condition of a Heegaard diagram implying that it is unstabilized. As applications, we show that Heegaard splittings of 2-fold branched coverings of n-component, n-bridge links in S3 are unstabilized, and we also construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.

scholarly journals ON CRITICAL HEEGAARD SPLITTINGS OF TUNNEL NUMBER TWO COMPOSITE KNOT EXTERIORS

2013 ◽  
Vol 22 (11) ◽  
pp. 1350065 ◽  
Author(s):  
JUNGSOO KIM
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Connected Sum ◽  
Tunnel Number

In this paper, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two (1,1)-knots always contains a critical Heegaard splitting in its exterior as the examples of the main theorem.

scholarly journals Many Haken Heegaard splittings

2018 ◽  
Vol 12 (02) ◽  
pp. 357-369
Author(s):  
Alessandro Sisto
Keyword(s):  
Hyperbolic Manifolds ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Simple Criterion ◽  
Heegaard Genus ◽  
Heegaard Distance ◽  
The Way

We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant. Along the way we give simpler and shorter proofs of the existence of splittings with specified Heegaard distance, originally proven by Ido–Jang–Kobayashi, of the existence of hyperbolic manifolds with prescribed Casson invariant, originally due to Lubotzky–Maher–Wu, and of a result about subsurface projections of disc sets (for which we even get better constants), originally due to Masur–Schleimer.

scholarly journals Haken spheres for genus two Heegaard splittings

2017 ◽  
Vol 165 (3) ◽  
pp. 563-572 ◽  
Author(s):  
SANGBUM CHO ◽  
YUYA KODA
Keyword(s):  
Heegaard Splitting ◽  
Combinatorial Structure ◽  
Lens Spaces ◽  
Heegaard Splittings ◽  
Precise Description ◽  
Connected Sum ◽  
Connected Sums ◽  
Genus 2 ◽  
Genus Two

AbstractA manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

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