The level sets of the chain length function derived from equilateral and equiangular 5-polygonal chains with simplicity

Abstract

We consider equilateral and equiangular $n$-polygonal chains in ${\mathbf{R}}^3$ with $n$ vertices, i.e. equilateral and equiangular spatial polygonal chains with $n$ vertices. Let $M_{\theta }(n)$ be the configuration space consisting of such polygonal chains with the bond angle $\theta$. In this article, we assume that $n=5$ and that $\theta$ satisfies $\frac{\pi }{2}\le \theta <\pi$. This bond angle condition gives that each polygonal chain in $M_{\theta }(5)$ is an equilateral and equiangular simple open or closed polygonal chain. The aim of this article is to study the level sets of the chain length function on $M_{\theta }(5)$ under the bond angle condition.