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In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.
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Knot data analysis using multiscale Gauss link integral
In the past decade, topological data analysis has emerged as a powerful algebraic topology approach in data science. 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 paradigm that incorporates 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 and offers a multiscale geometric topology approach to capture the local structures and connectivities in data. By integration with machine learning or deep learning, the proposed mGLI significantly outperforms other state-of-the-art methods across various benchmark problems in 13 intricately complex biological datasets, including protein flexibility analysis, protein–ligand interactions, human Ether-à-go-go-Related Gene potassium channel blockade screening, and quantitative toxicity assessment. Our KDA opens a research area—knot deep learning—in data science.
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- Award ID(s):
- 1900473
- PAR ID:
- 10566697
- Publisher / Repository:
- National Academy of Sciences
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 121
- Issue:
- 42
- ISSN:
- 0027-8424
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1073/pnas.2408431121
