Extremal polygons in R^3

TL;DR

This paper extends the study of Morse functions from planar to 3D polygonal linkages, showing that a generalized area function has better Morse properties in three dimensions, especially for equilateral polygons with an odd number of edges.

Contribution

It introduces a generalized area function for 3D polygonal linkages, demonstrating it is a perfect Morse function for certain cases, unlike in the planar scenario.

Findings

The generalized area function has an easy description of critical points in 3D.

For odd-edged equilateral polygons, the function is always a perfect Morse function.

Cyclic equilateral polygons generate homology groups of the configuration space.

Abstract

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ in many cases is not a perfect Morse function. In particular, for an equilateral pentagonal linkage it has one extra local maximum (except for the global maximum) and one extra local minimum. In the present paper we consider the space of 3D configurations of a polygonal linkage. For an appropriate generalization $S$ of the area function $A$ the situation becomes nicer: we again have an easy description of critical points and a simple formula for the Morse index. In particular, unlike the planar case, for an equilateral linkage with odd number of edges the function $S$ is always a perfect Morse function and fits the lacunary principle. Therefore cyclic equilateral polygons can be interpreted as independent generators of the homology groups of the (decorated) configuration space.