scholarly journals Incompressible surfaces and Dehn surgery on 1-bridge knots in handlebodies

1996 ◽  
Vol 120 (4) ◽  
pp. 687-696 ◽  
Author(s):  
Ying-Qing Wu
Keyword(s):  
Solid Torus ◽  
Exceptional Case ◽  
Isotopy Class ◽  
Geometric Method ◽  
Dehn Surgery ◽  
Closed Curves ◽  
Incompressible Surfaces ◽  
Regular Neighbourhood ◽  
Manifold Theory

Given a knot K in a 3-manifold M, we use N(K) to denote a regular neighbourhood of K. Suppose γ is a slope (i.e. an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H – Int N(K) so that γ bounds a meridional disc. We say that M is ∂-reducible if ∂M is compressible in M, and we call γ a ∂-reducing slope of K if (H, K; γ) is ∂-reducible. Since incompressible surfaces play an important rôle in 3-manifold theory, it is important to know what slopes of a given knot are ∂-reducing. In the generic case there are at most three ∂-reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂-reducing slopes only if it is a 1-bridge braid. Moreover, all such knots and their corresponding ∂-reducing slopes are classified in [1]. For 1-bridge braids with small bridge width, a geometric method of detecting ∂-reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e. if K is a knot in a handlebody with H – K ∂-irreducible, then K has ∂-reducing slopes only if K is a 1-bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems.

Dehn surgery and essential annuli

1996 ◽  
Vol 120 (1) ◽  
pp. 127-146 ◽  
Author(s):  
Chuichiro Hayashi
Keyword(s):  
Solid Torus ◽  
Dehn Surgery ◽  
The Core ◽  
Manifold With Boundary ◽  
Regular Neighbourhood ◽  
Intersection Points

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

4-Punctured Tori in the Exteriors of Knots

1997 ◽  
Vol 06 (05) ◽  
pp. 659-676 ◽  
Author(s):  
Mario Eudave-Muñoz
Keyword(s):  
Infinite Family ◽  
Solid Torus ◽  
Dehn Surgery ◽  
The Core

In this paper we construct an infinite family of hyperbolic knots, each having a Dehn surgery which produces a manifold containing an incompressible torus, which hits the core of the surgered solid torus in four points, but containing no incompressible torus hitting it in less than four points.

scholarly journals Normal closures of powers of Dehn twists in mapping class groups

1992 ◽  
Vol 34 (3) ◽  
pp. 314-317 ◽  
Author(s):  
Stephen P. Humphries
Keyword(s):  
Class Group ◽  
Simple Closed Curve ◽  
Isotopy Class ◽  
Dehn Twist ◽  
Closed Curve ◽  
Mapping Class ◽  
Class Groups ◽  
Closed Curves ◽  
Dehn Twists ◽  
Definition Of

LetF = F(g, n)be an oriented surface of genusg≥1withn<2boundary components and letM(F)be its mapping class group. ThenM(F)is generated by Dehn twists about a finite number of non-bounding simple closed curves inF([6, 5]). See [1] for the definition of a Dehn twist. Letebe a non-bounding simple closed curve inFand letEdenote the isotopy class of the Dehn twist aboute. LetNbe the normal closure ofE2inM(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:Theorem 1.The subgroup N is of finite index in M(F).

scholarly journals Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries

2020 ◽  
Vol 29 (11) ◽  
pp. 2050075
Author(s):  
Sungmo Kang
Keyword(s):  
Dehn Surgery ◽  
Surface Slope ◽  
Addition Formula ◽  
Closed Curves ◽  
Tunnel Number ◽  
Genus Two ◽  
Surgery Formula ◽  
Tunnel Number One Knots

Suppose [Formula: see text] and [Formula: see text] are disjoint simple closed curves in the boundary of a genus two handlebody [Formula: see text] such that [Formula: see text] (i.e. a 2-handle addition along [Formula: see text]) embeds in [Formula: see text] as the exterior of a hyperbolic knot [Formula: see text] (thus, [Formula: see text] is a tunnel-number-one knot), and [Formula: see text] is Seifert in [Formula: see text] (i.e. a 2-handle addition [Formula: see text] is a Seifert-fibered space) and not the meridian of [Formula: see text]. Then for a slope [Formula: see text] of [Formula: see text] represented by [Formula: see text], [Formula: see text]-Dehn surgery [Formula: see text] is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [J. Dean, Small Seifert-fibered Dehn surgery on hyperbolic knots, Algebr. Geom. Topol. 3 (2003) 435–472.]. In this paper, we show that there exists a meridional curve [Formula: see text] of [Formula: see text] (or [Formula: see text]) in [Formula: see text] such that [Formula: see text] intersects [Formula: see text] transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery [Formula: see text] can arise from a primitive/Seifert position of [Formula: see text] with [Formula: see text] its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in [Formula: see text] is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.

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.

scholarly journals Conjugacy invariants for Brouwer mapping classes

2016 ◽  
Vol 37 (6) ◽  
pp. 1765-1814
Author(s):  
JULIETTE BAVARD
Keyword(s):  
Isotopy Class ◽  
Closed Curves ◽  
Mapping Classes

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

scholarly journals Topological steps toward the Homflypt skein module of the lens spaces L(p,1) via braids

2016 ◽  
Vol 25 (14) ◽  
pp. 1650084 ◽  
Author(s):  
Ioannis Diamantis ◽  
Sofia Lambropoulou ◽  
Jozef H. Przytycki
Keyword(s):  
Infinite System ◽  
Solid Torus ◽  
Lens Spaces ◽  
System Of Equations ◽  
Closed Curves ◽  
Skein Modules ◽  
Skein Module ◽  
Basic Set ◽  
Knots And Links ◽  
Shed Light

In this paper, we work toward the Homflypt skein module of the lens spaces [Formula: see text], [Formula: see text] using braids. In particular, we establish the connection between [Formula: see text], the Homflypt skein module of the solid torus ST, and [Formula: see text] and arrive at an infinite system, whose solution corresponds to the computation of [Formula: see text]. We start from the Lambropoulou invariant [Formula: see text] for knots and links in ST, the universal analog of the Homflypt polynomial in ST, and a new basis, [Formula: see text], of [Formula: see text] presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra 220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014 , arXiv:1412.3642 [math.GT]]. We show that [Formula: see text] is obtained from [Formula: see text] by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis [Formula: see text], where the bbm are performed on any moving strand of each element in [Formula: see text]. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set [Formula: see text]. The importance of our approach is that it can shed light on the problem of computing skein modules of arbitrary c.c.o. [Formula: see text]-manifolds, since any [Formula: see text]-manifold can be obtained by surgery on [Formula: see text] along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.

MINIMISING THE BOUNDARIES OF PUNCTURED PROJECTIVE PLANES IN S3

2001 ◽  
Vol 10 (03) ◽  
pp. 415-430 ◽  
Author(s):  
MICHEL DOMERGUE ◽  
DANIEL MATIGNON
Keyword(s):  
Projective Plane ◽  
Solid Torus ◽  
Projective Planes ◽  
Dehn Surgery ◽  
The Core

This paper concerns 3-manifolds X obtained by Dehn surgery on a knot in S 3, in particular those which contain embedded projective planes. Either, they are homeomorphic to the 3-real projeclive space ℝ P 3, or they are reducible. Let p be the number of intersections of a projective plane in X with the core of the solid torus added during surgery. We prove here that either X is reducible or p is bigger than or equal to five. Consequently, if X is homeomorphic to ℝ P 3 then all its projective planes are pierced at least in five points by the core of the surgery. This result is considered as a step towards showing that ℝ P 3 cannot be obtained by a Dehn surgery along a knot in S 3.

scholarly journals Closed incompressible surfaces in 2-generator hyperbolic 3-manifolds with a single cusp

1993 ◽  
Vol 36 (3) ◽  
pp. 501-513
Author(s):  
D. D. Long ◽  
A. W. Reid
Keyword(s):  
Fundamental Group ◽  
Incompressible Surfaces ◽  
Essential Surface ◽  
Regular Neighbourhood ◽  
Genus 2 ◽  
Tunnel Number

A knot K is said to have tunnel number 1 if there is an embedded arc A in S3, with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K ∪ A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3.

Thicknesses of knots

1999 ◽  
Vol 126 (2) ◽  
pp. 293-310 ◽  
Author(s):  
Y. DIAO ◽  
C. ERNST ◽  
E. J. JANSE VAN RENSBURG
Keyword(s):  
Smooth Surface ◽  
Solid Torus ◽  
Great Care ◽  
Topological Properties ◽  
Closed Curves ◽  
Large Curvature ◽  
Basic Properties ◽  
The Core ◽  
Strong Deformation ◽  
The Relationship

In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.

Sign in / Sign up

Export Citation Format

Share Document