标题： 多尺度琼斯多项式和持久琼斯多项式用于结数据分析 显示英文标题

标题： Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis

摘要： 许多科学、工程和艺术中的结构可以被视为三维空间中的曲线。 这些曲线的纠缠在决定材料的功能性和物理性质中起着关键作用。 结理论中的许多概念为探索三维空间中曲线的复杂性和纠缠提供了理论工具。 然而，经典结理论主要关注全局拓扑性质，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。 在本工作中，提出了两种基于琼斯多项式的局部模型，即多尺度琼斯多项式和持续琼斯多项式。 分析了这些模型的稳定性，特别是多尺度和持续琼斯多项式模型对曲线集合中微小扰动的不敏感性，从而确保了它们在现实世界应用中的鲁棒性。 显示英文摘要

摘要： 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 primarily 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, namely the multiscale Jones polynomial and the persistent Jones polynomial, are proposed. The stability of these models, especially the insensitivity of the multiscale and persistent Jones polynomial models to small perturbations in curve collections, is analyzed, thus ensuring their robustness for real-world applications.