Evolutionary Khovanov homology

TL;DR

This paper introduces evolutionary Khovanov homology (EKH), a metric-based approach to knot theory that enables multiscale quantitative data analysis of real-world knots, revealing complex invariants at various scales.

Contribution

The work develops EKH, integrating metrics into knot theory for the first time, enhancing its applicability to data analysis and machine learning tasks.

Findings

EKH reveals non-trivial invariants at multiple scales.

EKH demonstrates potential for real-world data analysis.

EKH bridges qualitative topology and quantitative metrics.

Abstract

Knot theory is a study of the embedding of closed circles into three-dimensional Euclidean space, motivated the ubiquity of knots in daily life and human civilization. However, the current knot theory focuses on the topology rather than metric. As such, the application of knot theory remains primitive and qualitative. Motivated by the need of quantitative knot data analysis (KDA), this work implements the metric into knot theory, the evolutionary Khovanov homology (EKH), to facilitate a multiscale KDA of real-world data. It is demonstrated that EKH exhibits non-trivial knot invariants at appropriate scales even if the global topological structure of a knot is simple. The proposed EKH has a great potential for KDA and knot learning.