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.

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 ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

2020 ◽  
pp. 1-10
Author(s):  
MARK GRANT ◽  
AGATA SIENICKA
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Mapping Class ◽  
Closed Orientable Surface ◽  
Outer Action ◽  
Positive Genus ◽  
Closed Sphere ◽  
Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

Links with alternating diagrams on closed surfaces of positive genus

1995 ◽  
Vol 117 (1) ◽  
pp. 113-128 ◽  
Author(s):  
Chuichiro Hayashi
Keyword(s):  
Orientable Surface ◽  
Heegaard Splitting ◽  
Closed Orientable Surface ◽  
Closed Surfaces ◽  
Positive Genus

AbstractWe consider links in an orientable 3-manifold M which have an alternating diagram on a closed orientable surface F of positive genus. We see that if the diagram is ‘complex’ enough and if F gives a Heegaard splitting of M, then such a link L is prime and M—L does not contain an essential torus.

scholarly journals Dehn’s algorithm for simple diagrams

2015 ◽  
Vol 24 (14) ◽  
pp. 1550074
Author(s):  
Charles Frohman ◽  
Joanna Kania-Bartoszynska
Keyword(s):  
Conjugacy Class ◽  
Disjoint Union ◽  
Closed Surface ◽  
Intersection Numbers ◽  
Closed Curves ◽  
Class A ◽  
Positive Genus ◽  
Monotonically Decrease ◽  
Standard Presentation ◽  
Closed Oriented Surface

Dehn gave an algorithm for deciding if two cyclic words in the standard presentation of the fundamental group of a closed oriented surface of positive genus represent the same conjugacy class. A simple diagram on a surface is a disjoint union of simple closed curves none of which bound a disk. If [Formula: see text] is a once punctured closed surface of negative Euler characteristic, simple diagrams are classified up to isotopy by their geometric intersection numbers with the edges of an ideal triangulation of [Formula: see text]. Simple diagrams on the unpunctured surface [Formula: see text] can be represented by simple diagrams on [Formula: see text]. The weight of a simple diagram is the sum of its geometric intersection numbers with the edges of the triangulation. We show that you can pass from any representative to a least weight representative via a sequence of elementary moves, that monotonically decrease weights. This leads to a geometric analog of Dehn’s algorithm for simple diagrams.

scholarly journals The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes

2021 ◽  
Author(s):  
Yasuhiko Kamiyama
Keyword(s):  
Mathematical Model ◽  
Configuration Space ◽  
Topological Type ◽  
Closed Surface ◽  
Positive Genus

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.

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.

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

Holographic part integral and exemal closed surface

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Closed Surface

Holographic part integral and exemal closed surface

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