  In the past decade, topological data analysis (TDA) has emerged as a powerful
approach in data science. The main technique in TDA is persistent homology,
which tracks topological invariants over the filtration of point cloud data
using algebraic topology. Although knot theory and related subjects are a focus
of study in mathematics, their success in practical applications is quite
limited due to the lack of localization and quantization. We address these
challenges by introducing knot data analysis (KDA), a new paradigm that
incorporating curve segmentation and multiscale analysis into the Gauss link
integral. The resulting multiscale Gauss link integral (mGLI) recovers the
global topological properties of knots and links at an appropriate scale but
offers multiscale feature vectors to capture the local structures and
connectivities of each curve segment at various scales. The proposed mGLI
significantly outperforms other state-of-the-art methods in benchmark protein
flexibility analysis, including earlier persistent homology-based methods. Our
approach enables the integration of artificial intelligence (AI) and KDA for
general curve-like objects and data.
