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.
Keywords: 57K10; 92C10; Jones polynomial; curve data analysis; knot data analysis; localization; protein flexibility; stability.