Multi-scale Jones polynomial and persistent Jones polynomial for knot data analysis

Ruzhi Song; Fengling Li; Jie Wu; Fengchun Lei; Guo-Wei Wei; Ruzhi Song; Fengling Li; Jie Wu; Fengchun Lei; Guo-Wei Wei

doi:10.3934/math.2025068

AIMS Mathematics

2025, Volume 10, Issue 1: 1463-1487. doi: 10.3934/math.2025068

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Multi-scale Jones polynomial and persistent Jones polynomial for knot data analysis

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China
2.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, Hebei, China
3.
Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
4.
Department of Mathematics, Michigan State University, 426 Auditorium Road, East Lansing, MI 48824, USA
5.
Department of Biochemistry and Molecular Biology, Michigan State University, 426 Auditorium Road, East Lansing, MI 48824, USA
6.
Department of Electrical and Computer Engineering, Michigan State University, 426 Auditorium Road, East Lansing, MI 48824, USA

Received: 16 November 2024 Revised: 04 January 2025 Accepted: 13 January 2025 Published: 22 January 2025
MSC : 57K10, 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 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.
- knot data analysis,
- curve data analysis,
- Jones polynomial,
- localization,
- stability,
- protein flexibility
Citation: Ruzhi Song, Fengling Li, Jie Wu, Fengchun Lei, Guo-Wei Wei. Multi-scale Jones polynomial and persistent Jones polynomial for knot data analysis[J]. AIMS Mathematics, 2025, 10(1): 1463-1487. doi: 10.3934/math.2025068

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Abstract

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.

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