The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes

Mapping Intimacies ◽

10.5772/intechopen.100723 ◽

2021 ◽

Author(s):

Yasuhiko Kamiyama

Keyword(s):

Mathematical Model ◽

Configuration Space ◽

Topological Type ◽

Closed Surface ◽

Positive Genus

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.

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ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

Glasgow Mathematical Journal ◽

10.1017/s0017089520000208 ◽

2020 ◽

pp. 1-10

Author(s):

MARK GRANT ◽

AGATA SIENICKA

Keyword(s):

Braid Group ◽

Class Group ◽

Closed Surface ◽

Orientable Surface ◽

Mapping Class ◽

Closed Orientable Surface ◽

Outer Action ◽

Positive Genus ◽

Closed Sphere ◽

Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

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The Topological Type of Equilateral and Almost Equiangular Polygon Spaces

Journal of Mathematics ◽

10.1155/2019/6958218 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6

Author(s):

Yasuhiko Kamiyama

Keyword(s):

Configuration Space ◽

Topological Type ◽

Bond Angles

Let Pn be the configuration space of equilateral spatial n-gons. For θ∈0,π and k∈0,1,…,n, let Pnkθ be the subspace of Pn consisting of elements whose first k bond angles are θ. Recently, the topological type of Pnkθ was determined for small n, special θ, and k=n or n−2. In this paper, we determine the topological type of Pnn−3θ for general n and θ.

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The Configuration Space of n-Tuples of Equiangular Unit Vectors for n=3, 4, and 5

Chinese Journal of Mathematics ◽

10.1155/2018/9842324 ◽

2018 ◽

Vol 2018 ◽

pp. 1-4

Author(s):

Yasuhiko Kamiyama

Keyword(s):

Configuration Space ◽

Dimensional Space ◽

Topological Type ◽

Unit Vectors

Let Mn(θ) be the configuration space of n-tuples of unit vectors in R3 such that all interior angles are θ. The space Mn(θ) is an (n-3)-dimensional space. This paper determines the topological type of Mn(θ) for n=3, 4, and 5.

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Dehn’s algorithm for simple diagrams

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216515500741 ◽

2015 ◽

Vol 24 (14) ◽

pp. 1550074

Author(s):

Charles Frohman ◽

Joanna Kania-Bartoszynska

Keyword(s):

Conjugacy Class ◽

Disjoint Union ◽

Closed Surface ◽

Intersection Numbers ◽

Closed Curves ◽

Class A ◽

Positive Genus ◽

Monotonically Decrease ◽

Standard Presentation ◽

Closed Oriented Surface

Dehn gave an algorithm for deciding if two cyclic words in the standard presentation of the fundamental group of a closed oriented surface of positive genus represent the same conjugacy class. A simple diagram on a surface is a disjoint union of simple closed curves none of which bound a disk. If [Formula: see text] is a once punctured closed surface of negative Euler characteristic, simple diagrams are classified up to isotopy by their geometric intersection numbers with the edges of an ideal triangulation of [Formula: see text]. Simple diagrams on the unpunctured surface [Formula: see text] can be represented by simple diagrams on [Formula: see text]. The weight of a simple diagram is the sum of its geometric intersection numbers with the edges of the triangulation. We show that you can pass from any representative to a least weight representative via a sequence of elementary moves, that monotonically decrease weights. This leads to a geometric analog of Dehn’s algorithm for simple diagrams.

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Weakly reducible H′-splittings of 3-manifolds

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521400046 ◽

2021 ◽

Author(s):

Yaru Gao ◽

Fengling Li ◽

Liang Liang ◽

Fengchun Lei

Keyword(s):

Closed Surface ◽

Heegaard Splitting ◽

Planar Surface ◽

Connected Surface ◽

Positive Genus ◽

Weakly Reducible

We introduce the [Formula: see text]-splittings for 3-manifolds as follows. For a compact connected surface [Formula: see text] properly embedded in a compact connected orientable 3-manifold [Formula: see text], if [Formula: see text] decomposes [Formula: see text] into two handlebodies [Formula: see text] and [Formula: see text], then [Formula: see text] is called an [Formula: see text]-splitting for [Formula: see text]. Clearly, when [Formula: see text] is closed, this is just the Heegaard splitting for [Formula: see text]; when [Formula: see text] is with boundary, the [Formula: see text]-splitting for [Formula: see text] is different from the Heegaard splitting for [Formula: see text]. In this paper, we first show that any compact connected orientable 3-manifold admits an [Formula: see text]-splitting, then generalize Casson–Gordon theorem on weakly reducible Heegaard splitting to the [Formula: see text]-splitting case in the following version: if [Formula: see text] is a weakly reducible [Formula: see text]-splitting for a compact connected orientable 3-manifold [Formula: see text], then (1) [Formula: see text] contains an incompressible closed surface of positive genus or (2) the [Formula: see text]-splitting [Formula: see text] is reducible or (3) there is an essential 2-sphere [Formula: see text] in [Formula: see text] such that [Formula: see text] is a collection of essential disks in [Formula: see text] and [Formula: see text] is an incompressible and not boundary parallel planar surface in [Formula: see text] with at least two boundary components, where [Formula: see text] or (4) [Formula: see text] is stabilized.

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