Persistent Khovanov homology of tangles

TL;DR

This paper introduces persistent Khovanov homology for tangles, a new mathematical framework that captures local topological features in curve-type data, enhancing data analysis in geometric topology and data science.

Contribution

It develops a novel persistent Khovanov homology framework for tangles, including a concrete functor for computation and a planar algebra approach for practical applications.

Findings

Provides a functor mapping tangles to modules for homology computation

Constructs a tangle category without fixed boundaries using planar algebra

Establishes a theoretical foundation for topological data analysis of curves

Abstract

Knot data analysis (KDA), which studies data with curve-type structures such as knots, links, and tangles, has emerging as a promising geometric topology approach in data science. While evolutionary Khovanov homology has been developed to analyze the global persistent topological features of links, it has limitations in capturing the local topological characteristics of knots and links. To address this challenge, we introduce the persistent Khovanov homology of tangles, providing a new mathematical framework for characterizing local features in curve-type data. While tangle homology is inherently abstract, we provide a concrete functor which maps the category of tangles to the category of modules, enabling the computation of tangle homology. Additionally, we employ planar algebra to construct a category of tangles without invoking fixed boundaries, thereby giving rise to a persistent Khovanov homology functor that enables practical applications. This framework provides a theoretical foundation and practical strategies for the topological analysis of curve-type data.