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].

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].

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 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.

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 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.

scholarly journals Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

2014 ◽  
Vol 115 (2) ◽  
pp. 173 ◽  
Author(s):  
Fengling Li ◽  
Fengchun Lei ◽  
Guoqiu Yang
Keyword(s):  
Sufficient Conditions ◽  
Heegaard Splitting ◽  
Heegaard Distance

Let $M_{i}$ be a compact orientable 3-manifold, and $A_{i}$ an incompressible annulus on a component $F_i$ of $\partial M_i$, $i=1,2$. Suppose $A_{1}$ is separating on $F_{1}$ and $A_{2}$ is non-separating on $F_{2}$. Let $M$ be the annulus sum of $M_1$ and $M_2$ along $A_1$ and $A_2$. In the present paper we show that if $M_{i}$ has a Heegaard splitting $V_{i}\cup_{S_{i}}W_{i}$ with Heegaard distance $d(S_{i})\geq2g(M_{i})+5$ for $i=1,2$, then $g(M)=g(M_{1})+g(M_{2})$. Moreover, when $g(F_{2})\geq 2$, the minimal Heegaard splitting of $M$ is unique up to isotopy.

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

scholarly journals Distribution of points on Abelian covers over finite fields

2018 ◽  
Vol 14 (05) ◽  
pp. 1375-1401 ◽  
Author(s):  
Patrick Meisner
Keyword(s):  
Finite Fields ◽  
Galois Extension ◽  
Random Variables ◽  
Families Of Curves ◽  
Abelian Covers ◽  
Distribution Of Points ◽  
Genus Formula

We determine in this paper the distribution of the number of points on the covers of [Formula: see text] such that [Formula: see text] is a Galois extension and [Formula: see text] is abelian when [Formula: see text] is fixed and the genus, [Formula: see text], tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over [Formula: see text]. In all cases, the distribution is given by a sum of [Formula: see text] random variables.

Cuspidality in higher genus

2016 ◽  
Vol 12 (08) ◽  
pp. 2043-2060
Author(s):  
Dania Zantout
Keyword(s):  
Linear Operator ◽  
Half Space ◽  
Modular Forms ◽  
Cusp Forms ◽  
General Notion ◽  
Global Operator ◽  
Higher Genus ◽  
Genus Formula ◽  
Holomorphic Modular Forms

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.

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