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.

The amalgamation of high distance Heegaard splittings along tori

2019 ◽  
Vol 28 (02) ◽  
pp. 1950014
Author(s):  
Yang Li ◽  
Guoqiu Yang
Keyword(s):  
Disjoint Union ◽  
Heegaard Splitting ◽  
Heegaard Splittings ◽  
Heegaard Genus ◽  
Genus Formula ◽  
Distance Formula

Let [Formula: see text] be a compact, connected, orientable closed 3-manifold. Let [Formula: see text] be a disjoint union of incompressible separating tori in [Formula: see text] which cut [Formula: see text] into the submanifolds [Formula: see text]. In the present paper, we show that the Heegaard genus [Formula: see text] of [Formula: see text] is given by [Formula: see text] provided that each [Formula: see text] has a Heegaard splitting [Formula: see text] with Hempel distance [Formula: see text].

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.

On a hyperbolic 3-manifold with some special properties

1993 ◽  
Vol 113 (1) ◽  
pp. 87-90 ◽  
Author(s):  
B. Zimmermann
Keyword(s):  
Coxeter Groups ◽  
Universal Covering ◽  
Finite Subgroup ◽  
Heegaard Splitting ◽  
Heegaard Genus ◽  
Normal Torsion ◽  
Covering Group ◽  
Minimal Index ◽  
Torsion Free Subgroup ◽  
Torsion Free

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

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 Small 3-manifolds with large Heegaard distance

2013 ◽  
Vol 155 (3) ◽  
pp. 431-441 ◽  
Author(s):  
TAO LI
Keyword(s):  
Large Distance ◽  
Heegaard Splitting ◽  
Heegaard Distance

AbstractWe construct examples of closed non-Haken hyperbolic 3-manifolds with a Heegaard splitting of arbitrarily large distance.

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