Abstract

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.

Highlights

  • The configuration space of mechanical linkages in the Euclidean space of dimension three, known as polygon space, is the central objective in topological robotics

  • The linkage consists of n bars of length l1, ⋯, ln connected by revolving joints forming a closed spatial polygonal chain

  • The polygon space is quite important in various engineering applications: In molecular biology they describe varieties of molecular shapes, in robotics they appear as spaces of all possible configurations of some mechanisms, and they play a central role in statistical shape theory

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Summary

Introduction

The configuration space of mechanical linkages in the Euclidean space of dimension three, known as polygon space, is the central objective in topological robotics. The polygon space is quite important in various engineering applications: In molecular biology they describe varieties of molecular shapes, in robotics they appear as spaces of all possible configurations of some mechanisms, and they play a central role in statistical shape theory These spaces are very interesting: The symplectic structure on the polygon space was studied in the seminal paper [1]. Equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge Their configuration space is denoted by CnðθÞ. The mathematical model for cycloalkenes is the equilateral polygons whose interior angles are the same except for those of the both sides of the specified edge.

Main results
Proof of Theorem A
Proofs of Theorems B and C
Proof of Theorem D

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