Song Ruzhi, Li Fengling, Wu Jie, Lei Fengchun, Wei Guo-Wei
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China.
Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China.
AIMS Math. 2025;10(1):1463-1487. doi: 10.3934/math.2025068. Epub 2025 Jan 22.
Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial were proposed, namely, the multi-scale Jones polynomial and the persistent Jones polynomial. The stability of these models, especially the insensitivity of the multi-scale and persistent Jones polynomial models to small perturbations in curve collections, was analyzed, thus ensuring their robustness for real-world applications.
科学、工程和艺术中的许多结构都可以看作是三维空间中的曲线。这些曲线的缠结对确定材料的功能和物理性质起着至关重要的作用。纽结理论中的许多概念提供了理论工具，用于探索三维空间中曲线的复杂性和缠结。然而，经典纽结理论侧重于全局拓扑性质，缺乏对局部结构信息的考虑，而局部结构信息在实际应用中至关重要。在这项工作中，提出了两种基于琼斯多项式的局部化模型，即多尺度琼斯多项式和持久琼斯多项式。分析了这些模型的稳定性，特别是多尺度和持久琼斯多项式模型对曲线集合中小扰动的不敏感性，从而确保了它们在实际应用中的鲁棒性。
