Oriented area is a perfect Morse function

TL;DR

This paper demonstrates that a generalized oriented area function acts as a perfect Morse function on the configuration space of 3D equilateral polygons with an odd number of edges, linking Morse points to homology generators.

Contribution

It introduces a generalized oriented area function that serves as a perfect Morse function for certain polygonal configuration spaces, providing new topological insights.

Findings

Generalized oriented area function is a perfect Morse function.

Cyclic equilateral polygons correspond to Morse points and homology generators.

Results apply to 3D polygons with an odd number of edges.

Abstract

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generators of the homology groups of the (decorated) configuration space.