An equivalent description of the lens space L(p,q) with p prime

2020 ◽  
Vol 29 (10) ◽  
pp. 2042005
Author(s):  
Fengling Li ◽  
Dongxu Wang ◽  
Liang Liang ◽  
Fengchun Lei
Keyword(s):  
Complete System ◽  
Lens Space ◽  
Heegaard Splitting ◽  
Prime Integer ◽  
Heegaard Diagrams ◽  
Heegaard Diagram ◽  
Genus Formula ◽  
Equivalent Description

In the paper, we give an equivalent description of the lens space [Formula: see text] with [Formula: see text] prime in terms of any corresponding Heegaard diagrams as follows: Let [Formula: see text] be a closed orientable 3-manifold with [Formula: see text] and [Formula: see text] a Heegaard splitting of genus [Formula: see text] for [Formula: see text] with an associated Heegaard diagram [Formula: see text]. Assume [Formula: see text] is a prime integer. Then [Formula: see text] is homeomorphic to the lens space [Formula: see text] if and only if there exists an embedding [Formula: see text] such that [Formula: see text] bounds a complete system of surfaces for [Formula: see text].

scholarly journals CONSTRUCTING DOUBLY-POINTED HEEGAARD DIAGRAMS COMPATIBLE WITH (1,1) KNOTS

2013 ◽  
Vol 22 (11) ◽  
pp. 1350071
Author(s):  
PHILIP ORDING
Keyword(s):  
Normal Form ◽  
Floer Homology ◽  
Solid Torus ◽  
Heegaard Splitting ◽  
Heegaard Diagrams ◽  
Homology Groups ◽  
Knot Floer Homology ◽  
Heegaard Diagram ◽  
Train Tracks

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parametrization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This paper presents an algorithm for constructing a genus 1 doubly-pointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsváth and Szabó, provides a practical way to compute knot Floer homology groups for (1,1) knots. The construction uses train tracks, and its method is inspired by the work of Goda, Matsuda, and Morifuji.

DECOMPOSITIONS OF PLANAR HOMEOMORPHISMS USING COMPUTER SOFTWARE

2002 ◽  
Vol 11 (06) ◽  
pp. 955-972
Author(s):  
IL YEUN CHO ◽  
MITSUYUKI OCHIAI ◽  
YOSHIKO SAKATA
Keyword(s):  
Data Structure ◽  
Computer Software ◽  
The Self ◽  
Heegaard Diagrams ◽  
Heegaard Diagram ◽  
Connected Surface

We have established in [S3] an algorithm with a new data structure that decomposes gluing homeomorphisms of 3-manifolds given by planar Heegaard diagrams into a product of canonical Dehn's twists. To support this study, we developed a computer software called Decomposition of Planar Homeomorphisms (Genus 3) that automatically decomposes the self homeomorphis of a closed connected surface given by any planar Heegaard diagram of genus 3 into a product of canonical Dehn's twists. In this paper, we demonstrate the content and the implementation that this software holds and also show its availability.

scholarly journals An irreducible Heegaard diagram of the real projective 3-spacep3

2000 ◽  
Vol 23 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Young Ho Im ◽  
Soo Hwan Kim
Keyword(s):  
Projective Space ◽  
Real Projective Space ◽  
Heegaard Diagrams ◽  
The Real ◽  
Heegaard Diagram ◽  
Genus 2

We give a genus 3 Heegaard diagramHof the real projective spacep3, which has no waves and pairs of complementary handles. So Negami's result that every genus 2 Heegaard diagram ofp3is reducible cannot be extended to Heegaard diagrams ofp3with genus 3.

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.

A topologically minimal canonical Heegaard splitting of a surface bundle is critical

2018 ◽  
Vol 27 (05) ◽  
pp. 1850034
Author(s):  
Qiang E
Keyword(s):  
Topological Index ◽  
Topological Indices ◽  
Heegaard Splitting ◽  
Genus Formula ◽  
Heegaard Surfaces ◽  
Surface Bundle

Every surface bundle with genus [Formula: see text] fiber has a canonical Heegaard splitting of genus [Formula: see text]. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2.

scholarly journals Heegaard distance of the link complements in S3

2021 ◽  
pp. 2150005
Author(s):  
Xifeng Jin
Keyword(s):  
Heegaard Splitting ◽  
Link Complements ◽  
Genus Formula ◽  
Heegaard Distance ◽  
Distance Formula

We show that, for any integers, [Formula: see text] and [Formula: see text], there exists a link in [Formula: see text] such that its complement has a genus [Formula: see text] Heegaard splitting with distance [Formula: see text].

Properties of Casson–Gordon’s rectangle condition

2020 ◽  
Vol 29 (12) ◽  
pp. 2050083
Author(s):  
Bo-Hyun Kwon ◽  
Jung Hoon Lee
Keyword(s):  
Heegaard Splitting ◽  
Heegaard Diagram ◽  
Strongly Irreducible ◽  
Strong Irreducibility

For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

scholarly journals A note on irreducible Heegaard diagrams

2006 ◽  
Vol 2006 ◽  
pp. 1-11
Author(s):  
Alberto Cavicchioli ◽  
Fulvia Spaggiari
Keyword(s):  
Alternative Proof ◽  
Heegaard Diagrams ◽  
The Real ◽  
Connected Sums ◽  
Heegaard Diagram ◽  
Simple Alternative

We construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper.

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].

scholarly journals The Dual and Mirror Images of the Dunwoody 3-Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Soo Hwan Kim ◽  
Yangkok Kim
Keyword(s):  
Lens Space ◽  
Group Presentation ◽  
Heegaard Diagram

Recently, in 2013, we proved that certain presentations present the Dunwoody3-manifold groups. Since the Dunwoody3-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody3-manifolds. It is well known that every(1,1)-knots in a lens space can be represented by the set𝒟of the 4-tuples(a,b,c,r)(Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody3-manifolds, we proved the fact that the certain subset of𝒟representing all2-bridge knots of(1,1)-knots is determined completely by using the dual and mirror(1,1)-decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in𝒟as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).

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