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Multisacle Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis.

Ruzhi Song1,2, Fengling Li1, Jie Wu2

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

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|December 9, 2024
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Summary
This summary is machine-generated.

This study introduces localized models for analyzing curve entanglement, enhancing knot theory with local structural information for practical applications. The new multiscale and persistent Jones polynomial models offer robust analysis of complex curves.

Keywords:
57K1492C10Jones polynomialKnot data analysiscurve data analysislocalizationprotein flexibilitystability

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Area of Science:

  • Topology
  • Materials Science
  • Applied Mathematics

Background:

  • Complex curves in 3D space are fundamental in science, engineering, and art.
  • Curve entanglement significantly influences material properties and functionality.
  • Classical knot theory lacks local structural analysis, limiting practical applications.

Purpose of the Study:

  • To develop localized models for analyzing curve entanglement.
  • To incorporate local structural information into knot theory.
  • To enhance the practical applicability of knot theory.

Main Methods:

  • Proposed two localized models: the multiscale Jones polynomial and the persistent Jones polynomial.
  • Utilized concepts from knot theory and polynomial invariants.
  • Analyzed model stability against small perturbations.

Main Results:

  • Introduced the multiscale Jones polynomial for localized curve analysis.
  • Developed the persistent Jones polynomial, also focusing on local properties.
  • Demonstrated the stability and robustness of these new models.

Conclusions:

  • Localized models provide critical local structural information for curve entanglement.
  • The multiscale and persistent Jones polynomials offer robust tools for real-world applications.
  • These advancements bridge the gap between theoretical knot theory and practical structural analysis.