Morse theory and moduli spaces of self-avoiding polygonal linkages

2025/06/07 by Te Ba, Ze Zhou, Ba, Te +1
Engineering · Mathematics · #52C25 #57R70 #68U15 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.2506.06781

openalex publication_date 2025/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a smooth d-manifold M is diffeomorphic to \mathbb R^d if it admits a Lyapunov-Reeb function, i.e., a smooth map f:M→\mathbb R that is proper, lower-bounded, and has a unique critical point. By constructing such functions, we prove that the moduli spaces of self-avoiding polygonal linkages and configurations are diffeomorphic to Euclidean spaces. This resolves the Refined Carpenter's Rule Problem and confirms a conjecture proposed by González and Sedano-Mendoza. Furthermore, we describe foliation structures of these moduli spaces via level sets of Lyapunov-Reeb functions and develop algorithms for related problems.

Citations

• The space of immersed polygons
• Moduli space of simple polygons
• Polygons inscribed in Jordan curves with prescribed edge ratios
• Intersection homology of linkage spaces
• The topology of spaces of polygons
• Configuration spaces of convex and embedded polygons in the plane
• On the quantization of polygon spaces
• Homology of planar polygon spaces
• Homology of planar polygon spaces
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ricci flow with surgery on three-manifolds
• The entropy formula for the Ricci flow and its geometric applications
• Long Proteins with Unique Optimal Foldings in the H-P Model
• Universality theorems for configuration spaces of planar linkages
• Generalized delaunay triangulation for planar graphs

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• Discrete Morse Theory Is At Least As Perfect As Morse Theory
• Smoothing discrete Morse theory