Persistent Khovanov homology of tangles

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.