Morse theory and moduli spaces of self-avoiding polygonal linkages

Te Ba; Ze Zhou

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arXiv · 2506.06781

Morse theory and moduli spaces of self-avoiding polygonal linkages

Te Ba·Ze Zhou

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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 $f:M\to\mathbb R$ 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 Gonz\'{a}lez 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.

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Te Ba, Ze Zhou. 2025-06-07. Morse theory and moduli spaces of self-avoiding polygonal linkages. https://arxiv.org/abs/2506.06781

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