Evolutionary Khovanov homology

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.