The closed chains with spherical configuration spaces

Abstract

As a mathematical model of $n$-membered ringed hydrocarbon molecules, we consider closed chains in $\R^{3}$. Assume that the bond angle $\theta$ satisfies $\frac{n-4}{n-2}\pi0\theta0\frac{n-2}{n}\pi$ when $n=5,6,7$, and that $\frac{5}{7}\pi \leq \theta0 \frac{3}{4}\pi$ when $n=8$. Then the configuration space $C_{n}$ of the model is homeomorphic to $(n-4)$-dimensional sphere $S^{n-4}$. By this result, it is possible for approximating larger macrocyclic molecules by smaller ones to be more widely applied.