Closed braids which are not prime knots

1979 ◽  
Vol 86 (3) ◽  
pp. 421-426 ◽  
Author(s):  
H. R. Morton
Keyword(s):  
Braid Group ◽  
Connected Sum ◽  
Image Position

An element B ∈ Bn, the braid group on n strings, which can be written asin terms of the standard generators of Bn is called a split braid. It is easy to see that the resulting closed braid is the connected sum of the closed braids Ĉ and on k + 1 and n − k strings respectively (figure 1).

Knot surgery and primeness

1986 ◽  
Vol 99 (1) ◽  
pp. 89-102 ◽  
Author(s):  
Francisco González Acuña ◽  
Hamish Short
Keyword(s):  
Dehn Surgery ◽  
Connected Sum ◽  
Low Dimensional Topology ◽  
Knot Surgery ◽  
Low Dimensional ◽  
Image Position

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decompositionone of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

A toral configuration space and regular semisimple conjugacy classes

1995 ◽  
Vol 118 (1) ◽  
pp. 105-113 ◽  
Author(s):  
G. I. Lehrer
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Cohomology Ring ◽  
General Setting ◽  
De Rham Cohomology ◽  
Pure Braid Group ◽  
De Rham ◽  
Image Position

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration spaceThe symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

scholarly journals On reducible braids and composite braids

1994 ◽  
Vol 36 (2) ◽  
pp. 197-199
Author(s):  
Stephen P. Humphries
Keyword(s):  
Braid Group ◽  
Image Position

The braid group on n strings Bn has a presentation as a group with generators σ1, …, σn−1 and relations

Towards a characterization of smooth braids

1982 ◽  
Vol 92 (3) ◽  
pp. 425-427 ◽  
Author(s):  
D. L. Johnson
Keyword(s):  
Braid Group ◽  
Trivial Group ◽  
Single Crossing ◽  
Image Position

The braid group on n + 1 strings has the presentationwhere xi corresponds to the single crossing of the i string over the i + 1 string, and rij is a commutator when i < j − 1, and a braid relator when i = j − 1. B1 is the trivial group and B2 is infinite cyclic; we shall assume n ≥ 2 throughout.

scholarly journals How do autodiffeomorphisms act on embeddings?

2017 ◽  
Vol 148 (4) ◽  
pp. 835-848
Author(s):  
A. Skopenkov
Keyword(s):  
Homology Class ◽  
Connected Sum ◽  
Smooth Category ◽  
Image Position

We work in the smooth category. The following problem was suggested by E. Rees in 2002: describe the precomposition action of self-diffeomorphisms of Sp × Sq on the set of isotopy classes of embeddings Sp × Sq → ℝm.Let G: Sp × Sq → ℝm be an embedding such thatis null-homotopic for some pair of different points a, b ∈ Sp. We prove the following statement: if ψ is an autodiffeomorphism of Sp × Sq identical on a neighbourhood of a × Sq for some a ∈ Sp and p ⩽ q and 2m ⩾ 3p +3q + 4, then G◦ ψ is isotopic to G.Let N be an oriented (p + q)-manifold and let f, g be isotopy classes of embeddings N → ℝm, Sp × Sq → ℝm, respectively. As a corollary we obtain that under certain conditions for orientation-preserving embeddings s: Sp × Dq → N the Sp-parametric embedded connected sum f#sg depends only on f, g and the homology class of s|Sp × 0.

On 3-Manifolds with Sufficiently Large Decompositions

1971 ◽  
Vol 23 (4) ◽  
pp. 746-748
Author(s):  
Wolfgang Heil
Keyword(s):  
Fibre Bundle ◽  
Fundamental Groups ◽  
Connected Sum ◽  
Peripheral Structure ◽  
Image Position

In [6] it is shown that two (compact) orientable 3-manifolds which are irreducible, boundary irreducible and sufficiently large are homeomorphic if and only if there exists an isomorphism between the fundamental groups which respects the peripheral structure. In this note we extend this theorem to reducible 3-manifolds.Any compact 3-manifold M has a decomposition into prime manifolds [1; 4].1Here the connected sum of two bounded manifolds N1, N2 is denned by removing 3-balls B1 B2 in int N1, int N2, respectively, and glueing the resulting boundary spheres together. The M1's which occur in the decomposition (1) are either irreducible or handles (i.e., a fibre bundle over S1 with fibre S2). If (1) contains a fake 3-sphere, we assume it to be Mn.

scholarly journals Analogues of the braid group whose graphs are stars

1985 ◽  
Vol 28 (1) ◽  
pp. 35-39 ◽  
Author(s):  
Andrew K. Napthine
Keyword(s):  
Braid Group ◽  
Image Position

Let G be a group having presentationa braid relator and a commutator respectively, and define the graph of such a group as having vertices labelled x1,…, xn + 1 and such that xi and xj are joined if and only if rij = (xi, xj).

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

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