2-STRING FREE TANGLES AND INCOMPRESSIBLE SURFACES

2009 ◽  
Vol 18 (08) ◽  
pp. 1081-1087 ◽  
Author(s):  
YANNAN LI
Keyword(s):  
Positive Integer ◽  
Incompressible Surface ◽  
Connected Sum ◽  
Incompressible Surfaces

Suppose k is a connected sum of two knots, one of which admits a 2-string essential free tangle decomposition, then the exterior of k contains an incompressible surface of genus n for each positive integer n.

scholarly journals Invariant incompressible surfaces in reducible 3-manifolds

2018 ◽  
Vol 39 (11) ◽  
pp. 3136-3143 ◽  
Author(s):  
CHRISTOFOROS NEOFYTIDIS ◽  
SHICHENG WANG
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Incompressible Surface ◽  
Mapping Torus ◽  
Incompressible Surfaces ◽  
Hyperbolic Automorphism

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$ . As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$ , or the mapping torus of a hyperbolic automorphism.

Arcbodies

1983 ◽  
Vol 94 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Maria Teresa Lozano
Keyword(s):  
Recent Work ◽  
Incompressible Surface ◽  
Incompressible Surfaces ◽  
Positive Genus

A Haken manifold is a compact, orientable, irreducible 3-manifold which contains a properly embedded 2-sided, incompressible surface of positive genus. These manifolds are important in connection with the work of Haken, Waldhausen and the more recent work of Thurston (8). Thus it is interesting to investigate criteria for testing incompressible surfaces on 3-manifolds.

scholarly journals INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

2006 ◽  
Vol 15 (07) ◽  
pp. 935-948 ◽  
Author(s):  
MARIO EUDAVE-MUÑOZ
Keyword(s):  
Lens Space ◽  
Regular Neighborhood ◽  
Incompressible Surface ◽  
Special Position ◽  
Incompressible Surfaces ◽  
Toroidal Graph ◽  
Tunnel Number ◽  
Tunnel Number One Knots

Let M be S3, S1 × S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.

scholarly journals Diffeomorphism groups of connected sum of a product of spheres and classification of manifolds

1987 ◽  
Vol 10 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Samuel Omoloye Ajala
Keyword(s):  
Positive Integer ◽  
Complete Classification ◽  
Diffeomorphism Groups ◽  
Connected Sum ◽  
Homology Groups ◽  
Product Of Spheres

In [1] and [2] a classification of a manifoldMof the type(n,p,1)was given whereHp(M)=Hn−p(M)=ℤis the only non-trivial homology groups. In this paper we give a complete classification of manifolds of the type(n,p,2)and we extend the result to manifolds of type(n,p,r)whereris any positive integer andp=3,5,6,7mod(8).

Accidental surfaces in knot complements

2000 ◽  
Vol 09 (06) ◽  
pp. 725-733 ◽  
Author(s):  
Kazuhiro Ichihara ◽  
Makoto Ozawa
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Incompressible Surface ◽  
Incompressible Surfaces ◽  
Parallel Surface ◽  
Seifert Surfaces ◽  
Knot Complements ◽  
Necessary And Sufficient

It is well known that for many knot classes in the 3-sphere, every closed incompressible surface in their complements contains an essential loop which is isotopic into the boundary of the knot exterior. In this paper, we investigate closed incompressible surfaces in knot complements with this property. We show that if a closed, incompressible, non-boundary-parallel surface in a knot complement has such loops, then they determine the unique slope on the boundary of the knot exterior. Moreover, if the slope is non-meridional, then such loops are mutually isotopic in the surface. As an application, a necessary and sufficient condition for knots to bound totally knotted Seifert surfaces is given.

Centralizers of reflections in crystallographic groups

1982 ◽  
Vol 92 (1) ◽  
pp. 79-91 ◽  
Author(s):  
T. Baskan ◽  
A. M. Macbeath
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Plane ◽  
Universal Covering ◽  
Interesting Case ◽  
Discrete Groups ◽  
Incompressible Surface ◽  
Crystallographic Groups ◽  
Incompressible Surfaces ◽  
Universal Covering Space ◽  
Manifold Theory

The study of hyperbolic 3-manifolds has recently been recognized as an increasingly important part of 3-manifold theory (see (9)) and for some time the presence of incompressible surfaces in a 3-manifold has been known to be important (see, for example, (4)). A particularly interesting case occurs when the incompressible surface unfolds in the universal covering space into a hyperbolic plane. The fundamental group of the surface is then contained in the stabilizer of the plane, or, what is the same thing, in the centralizer of the reflection defined by the plane. This is one motivation for studying centralizers of reflections in discrete groups of hyperbolic isometries, or, as we shall call them, hyperbolic crystallographic groups.

Homology of groups of surfaces in the 4-sphere

1981 ◽  
Vol 89 (1) ◽  
pp. 113-117 ◽  
Author(s):  
C. McA. Gordon
Keyword(s):  
Positive Integer ◽  
Fundamental Group ◽  
Orientable Surface ◽  
Connected Sum ◽  
Connected Sums ◽  
Ambient Isotopy

1. Let F be a closed, connected, orientable surface of genus g ≥ 0 smoothly embedded in S4, and let π denote the fundamental group π1(S4 − F). Then H2(π) is a quotient of H2(S4 − F) ≅ H1(F) ≅ Z2g. If F is unknotted, that is, if there is an ambient isotopy taking F to the standardly embedded surface of genus g in S3 ⊂ S4, then π ≅ Z, so H2(π) = 0. More generally, if F is the connected sum of an unknotted surface and some 2-sphere S, then π ≅ π1 (S4 − S), so again H2(π) = 0. The question of whether H2(π) could ever be non-zero was raised in (5), Problem 4.29, and (10), Conjecture 4.13, and answered in (7) and (1). There, surfaces are constructed with H2(π)≅ Z/2, and hence, by forming connected sums, with H2(π) ≅ (Z/2)n for any positive integer n. In fact, (1) produces tori T in S4 with H2(π) ≅ Z/2, and hence surfaces of genus g with H2(π) ≅ (Z/2)g.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

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