Discrete Morse theory for moduli spaces of flexible polygons, or solitaire game on the circle

TL;DR

This paper develops a perfect discrete Morse function on the moduli space of polygonal linkages, simplifying its cell structure to facilitate topological analysis.

Contribution

It introduces a novel discrete Morse function tailored for the moduli space of flexible polygons, optimizing cell reduction using discrete Morse theory.

Findings

Minimal cell structure achieved for the moduli space

Enhanced understanding of the topology of polygonal linkage spaces

Method applicable to other moduli spaces with cell structures

Abstract

We introduce a perfect discrete Morse function on the moduli space of a polygonal linkage. The ingredients of the construction are: (1) the cell structure on the moduli space, and (2) the discrete Morse theory approach, which allows to reduce the number of cells to the minimal possible.