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.

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.

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.

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 THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

2013 ◽  
Vol 12 (06) ◽  
pp. 1350012 ◽  
Author(s):  
JAWAD ABUHLAIL ◽  
CHRISTIAN LOMP
Keyword(s):  
Commutative Ring ◽  
Theoretical Approach ◽  
Associative Ring ◽  
Ring Theory ◽  
Irreducible Elements ◽  
Strongly Irreducible ◽  
Torsion Theories ◽  
Commutative Ring Theory ◽  
Strong Irreducibility

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

INTERSECTIONS OF CURVES ON SURFACES WITH DISK FAMILIES IN HANDLEBODIES

2006 ◽  
Vol 15 (05) ◽  
pp. 631-649 ◽  
Author(s):  
JOEL ZABLOW
Keyword(s):  
Coordinate System ◽  
Dehn Twist ◽  
Heegaard Splitting ◽  
Closed Curves ◽  
Dehn Twists ◽  
Sufficiency Condition ◽  
Strongly Irreducible ◽  
Curves On Surfaces ◽  
Counting Formula

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection [Formula: see text] as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

Finite Projective Planes that Admit a Strongly Irreducible Collineation Group

1985 ◽  
Vol 37 (4) ◽  
pp. 579-611 ◽  
Author(s):  
Chat Yin Ho
Keyword(s):  
Group Theory ◽  
Coding Theory ◽  
Collineation Group ◽  
Incidence Matrix ◽  
Minimal Normal Subgroup ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Strongly Irreducible ◽  
Point Line ◽  
Strong Irreducibility

This paper studies how coding theory and group theory can be used to produce information about a finite projective plane π and a collineation group G of π.A new proof for Hering's bound on |G| is given in 2.5. Using the idea of coding theory developed in [9], a relation between two rows of the incidence matrix of π with respect to a tactical decomposition is obtained in 2.1. This result yields, among other things, some techniques in calculating |G|, and generalizes a result of Roth [16], [see 2.4 and 2.5].Hering [7] introduced the notion of strong irreducibility of G, that is, G does not leave invariant any point, line, triangle or proper subplane. He showed that if in addition G contains a non-trivial perspectivity, then there is a unique minimal normal subgroup of G. This subgroup is either non-abelian simple or isomorphic to the elementary abelian group Z3 × Z3 of order 9.

THE STRONG IRREDUCIBILITY OF A CLASS OF COWEN–DOUGLAS OPERATORS ON BANACH SPACES

2016 ◽  
Vol 94 (3) ◽  
pp. 479-488
Author(s):  
LIQIONG LIN ◽  
YUNNAN ZHANG
Keyword(s):  
Banach Spaces ◽  
Open Subset ◽  
Hilbert Spaces ◽  
Strongly Irreducible ◽  
Connected Open Subset ◽  
Strong Irreducibility

Let ${\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ be the set of Cowen–Douglas operators of index $n$ on a nonempty bounded connected open subset $\unicode[STIX]{x1D6FA}$ of $\mathbb{C}$. We consider the strong irreducibility of a class of Cowen–Douglas operators ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ on Banach spaces. We show ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})\subseteq {\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ and give some conditions under which an operator $T\in {\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ is strongly irreducible. All these results generalise similar results on Hilbert spaces.

Sign in / Sign up

Export Citation Format

Share Document