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

Ruzhi Song1,2, Fengling Li1, Jie Wu3,2

  • 1School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China.

AIMS Mathematics
|August 21, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces localized knot theory models, the multi-scale and persistent Jones polynomials, to analyze curve entanglement. These robust models capture local structural details crucial for material properties and real-world applications.

Keywords:
57K1092C10Jones polynomialcurve data analysisknot data analysislocalizationprotein flexibilitystability

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Area of Science:

  • * Interdisciplinary applications spanning science, engineering, and art.
  • * Utilizes concepts from knot theory for analyzing 3D curves.

Background:

  • * Curve entanglement is vital for material functionality and physical properties.
  • * Classical knot theory lacks local structural information critical for practical uses.

Purpose of the Study:

  • * To develop localized models for analyzing curve entanglement in 3-space.
  • * To address limitations of classical knot theory by incorporating local structural details.

Main Methods:

  • * Proposed two localized models: the multi-scale Jones polynomial and the persistent Jones polynomial.
  • * Analyzed the stability and robustness of these novel models.

Main Results:

  • * Developed localized Jones polynomial models capturing local curve features.
  • * Demonstrated model stability and insensitivity to minor perturbations in curve data.

Conclusions:

  • * The multi-scale and persistent Jones polynomials offer robust tools for analyzing complex curve entanglement.
  • * These localized models enhance the practical applicability of knot theory in real-world scenarios.