Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910716644032512 |
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| author | Song, Ruzhi Li, Fengling Wu, Jie Lei, Fengchun Wei, Guo-Wei |
| author_facet | Song, Ruzhi Li, Fengling Wu, Jie Lei, Fengchun Wei, Guo-Wei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17331 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis Song, Ruzhi Li, Fengling Wu, Jie Lei, Fengchun Wei, Guo-Wei Geometric Topology Biomolecules 57K14, 92C10 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. |
| title | Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis |
| topic | Geometric Topology Biomolecules 57K14, 92C10 |
| url | https://arxiv.org/abs/2411.17331 |