Bicycle tracks with hyperbolic monodromy -- results and conjectures

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Summary

The article discusses the bicycle monodromy of plane curves, specifically the conditions for a curve to have hyperbolic monodromy. The authors provide examples of rectangles and ellipses to illustrate the concepts and prove two theorems: Theorem 2, which states that a curve with curvature |κ| ≤ 1 has hyperbolic monodromy, and Theorem 3, which states that a convex curve with hyperbolic monodromy has length L > 2π.

Highlights

• The bicycle monodromy of a plane curve is a Möbius transformation that depends on the curve's geometry.
• The authors prove Theorem 2, which provides a sufficient condition for a curve to have hyperbolic monodromy.
• The authors prove Theorem 3, which provides a necessary condition for a convex curve to have hyperbolic monodromy.
• The article discusses the Arm Lemma, which is used to prove Theorem 3.
• The authors provide examples of rectangles and ellipses to illustrate the concepts.
• The article mentions the Menzin Conjecture, which states that a curve with area A > π has hyperbolic monodromy.
• The authors discuss the relation between the rotation numbers of the rear and front tracks of a bicycle.

Key Insights

• The bicycle monodromy of a plane curve is a fundamental concept in geometry, and understanding its properties is crucial for various applications.
• Theorem 2 provides a sufficient condition for a curve to have hyperbolic monodromy, which is essential for understanding the behavior of curves under certain transformations.
• Theorem 3 provides a necessary condition for a convex curve to have hyperbolic monodromy, which helps to narrow down the possibilities for curves with this property.
• The Arm Lemma is a crucial tool for proving Theorem 3, and its proof relies on a deep understanding of the geometry of curves.
• The examples of rectangles and ellipses illustrate the concepts and provide insight into the behavior of curves with different geometries.
• The Menzin Conjecture is an open problem that has been studied extensively, and its resolution would have significant implications for the field of geometry.
• The relation between the rotation numbers of the rear and front tracks of a bicycle is a fundamental concept in the geometry of curves, and understanding this relation is essential for various applications.

Mindmap

Citation

Bor, G., Hernández-Lamoneda, L., & Tabachnikov, S. (2024). Bicycle tracks with hyperbolic monodromy -- results and conjectures (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18676