Piecewise-linear-category approach to entanglement

1984 ◽  
Vol 3 (1) ◽  
pp. 121-136
Author(s):  
C. Agnes ◽  
M. Rasetti
Keyword(s):  
Piecewise Linear ◽  
Linear Category

Totally knotted knots are prime

1982 ◽  
Vol 91 (3) ◽  
pp. 467-472
Author(s):  
J. C. Gomez-Larran¯aga
Keyword(s):  
Finite Number ◽  
Knot Theory ◽  
Piecewise Linear ◽  
Connected Sum ◽  
Higher Dimensional ◽  
The Family ◽  
Linear Category

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

On braid monodromies of non-simple braided surfaces

1996 ◽  
Vol 120 (2) ◽  
pp. 237-245 ◽  
Author(s):  
Seiichi Kamada
Keyword(s):  
Knot Theory ◽  
Piecewise Linear ◽  
Tubular Neighbourhood ◽  
Oriented Surface ◽  
Covering Map ◽  
Branched Covering ◽  
Ambient Isotopy ◽  
Linear Category ◽  
Similar Notion ◽  
Closed Oriented Surface

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that and pr2 │F: F → S2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

On The 4-Dimensional Poincarè Conjecture for Manifolds with 2-Dimensional Spines

1974 ◽  
Vol 17 (4) ◽  
pp. 549-552
Author(s):  
R. P. Osborne
Keyword(s):  
Piecewise Linear ◽  
Connected Sum ◽  
Poincare Conjecture ◽  
Linear Category

We shall work in the piecewise-linear category, so that all manifolds and subsets thereof, as well as all maps are assumed to be piecewise-linear. If M is a manifold, denote by #kM the k-fold connected sum of copies of M and by 2M the double of M, that is the manifold obtained by sewing two copies of M together by the identity map on their boundaries.

First Homology of Irreducible 3-Manifolds

1971 ◽  
Vol 23 (4) ◽  
pp. 686-691 ◽  
Author(s):  
Benny Evans
Keyword(s):  
Abelian Group ◽  
Homology Group ◽  
Piecewise Linear ◽  
Finitely Generated ◽  
Linear Subspaces ◽  
Infinite Collection ◽  
Linear Category

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.

scholarly journals Manifold Neighbourhoods and a Conjecture of Adjamagbo

10.53733/131 ◽  
2021 ◽  
Vol 52 ◽  
pp. 167-174
Author(s):  
David Gauld
Keyword(s):  
Compact Subset ◽  
Piecewise Linear ◽  
Topological Category ◽  
Neighbourhood Basis ◽  
Linear Category

We verify a conjecture of P. Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set. We use smooth collars in the differential category, regular neighbourhoods in the piecewise linear category and handlebodies in the topological category.

Chaos in a manifold piecewise linear system

1981 ◽  
Vol 64 (10) ◽  
pp. 9-17 ◽  
Author(s):  
Toshimichi Saito ◽  
Hiroichi Fujita
Keyword(s):  
Linear System ◽  
Piecewise Linear ◽  
Piecewise Linear System
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