CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

2004 ◽  
Vol 13 (06) ◽  
pp. 829-843 ◽  
Author(s):  
YOAV MORIAH ◽  
ERIC SEDGWICK
Keyword(s):  
Boundary Component ◽  
Closed Surface ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Manifolds With Boundary ◽  
Minimal Genus ◽  
Essential Surfaces ◽  
Two Component ◽  
Single Boundary ◽  
Weakly Reducible

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

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.

Weakly reducible H′-splittings of 3-manifolds

2021 ◽  
Author(s):  
Yaru Gao ◽  
Fengling Li ◽  
Liang Liang ◽  
Fengchun Lei
Keyword(s):  
Closed Surface ◽  
Heegaard Splitting ◽  
Planar Surface ◽  
Connected Surface ◽  
Positive Genus ◽  
Weakly Reducible

We introduce the [Formula: see text]-splittings for 3-manifolds as follows. For a compact connected surface [Formula: see text] properly embedded in a compact connected orientable 3-manifold [Formula: see text], if [Formula: see text] decomposes [Formula: see text] into two handlebodies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an [Formula: see text]-splitting for [Formula: see text]. Clearly, when [Formula: see text] is closed, this is just the Heegaard splitting for [Formula: see text]; when [Formula: see text] is with boundary, the [Formula: see text]-splitting for [Formula: see text] is different from the Heegaard splitting for [Formula: see text]. In this paper, we first show that any compact connected orientable 3-manifold admits an [Formula: see text]-splitting, then generalize Casson–Gordon theorem on weakly reducible Heegaard splitting to the [Formula: see text]-splitting case in the following version: if [Formula: see text] is a weakly reducible [Formula: see text]-splitting for a compact connected orientable 3-manifold [Formula: see text], then (1) [Formula: see text] contains an incompressible closed surface of positive genus or (2) the [Formula: see text]-splitting [Formula: see text] is reducible or (3) there is an essential 2-sphere [Formula: see text] in [Formula: see text] such that [Formula: see text] is a collection of essential disks in [Formula: see text] and [Formula: see text] is an incompressible and not boundary parallel planar surface in [Formula: see text] with at least two boundary components, where [Formula: see text] or (4) [Formula: see text] is stabilized.

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.

Combinatorial Local Planarity and the Width of Graph Embeddings

1992 ◽  
Vol 44 (6) ◽  
pp. 1272-1288 ◽  
Author(s):  
Bojan Mohar
Keyword(s):  
Planar Graphs ◽  
Closed Surface ◽  
Graph Embeddings ◽  
Minimal Genus ◽  
Large Neighbourhood ◽  
Simply Connected

AbstractLet G be a graph embedded in a closed surface. The embedding is “locally planar” if for each face, a “large” neighbourhood of this face is simply connected. This notion is formalized, following [RV], by introducing the width ρ(ψ) of the embedding ψ. It is shown that embeddings with ρ(ψ) ≥ 3 behave very much like the embeddings of planar graphs in the 2-sphere. Another notion, “combinatorial local planarity”, is introduced. The criterion is independent of embeddings of the graph, but it guarantees that a given cycle in a graph G must be contractible in any minimal genus embedding of G (either orientable, or non-orientable). It generalizes the width introduced before. As application, short proofs of some important recently discovered results about embeddings of graphs are given and generalized or improved. Uniqueness and switching equivalence of graphs embedded in a fixed surface are also considered.

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.

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