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Evolutionary Khovanov homology.

Li Shen1, Jian Liu2,1, Guo-Wei Wei1,3,4

  • 1Department of Mathematics, Michigan State University, MI 48824, USA.

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|September 29, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces evolutionary Khovanov homology (EKH) for quantitative knot data analysis. EKH enables multiscale analysis of complex knot configurations, revealing hidden topological features for advanced applications.

Keywords:
55N3157K1057K18Khovanov homologygeometric topologyknotlinkmultiscalepersistent Khovanov topology

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Area of Science:

  • Geometric Topology
  • Computational Topology

Background:

  • Knot theory traditionally lacks metric analysis, limiting applications to qualitative insights.
  • Real-world knot data analysis requires quantitative methods beyond classical topological invariants.

Purpose of the Study:

  • To implement evolutionary Khovanov homology (EKH) for multiscale quantitative knot data analysis (KDA).
  • To capture multiscale topological features of knot configurations using specific link filtering metrics.

Main Methods:

  • Implementation of evolutionary Khovanov homology (EKH).
  • Application of EKH to filter links and analyze multiscale topological features.
  • Comparison with traditional knot invariants and other data analysis forms.

Main Results:

  • EKH successfully facilitates multiscale KDA of real-world knot data.
  • EKH reveals non-trivial knot invariants at specific scales, even for simple global knot structures.
  • Demonstrated capability to capture complex topological features beyond traditional methods.

Conclusions:

  • The proposed EKH offers a powerful tool for quantitative knot data analysis.
  • EKH has significant potential for machine learning applications involving knot-type data.
  • EKH provides a novel approach for analyzing complex topological data structures.