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Data Driven Perspectives on Knot Theory

Pawel Dlotko, Davide Gurnari, Radmila Sazdanovic

TL;DR

The paper argues that knot invariants can be meaningfully studied as high-dimensional data, using topological data analysis tools to reveal structure and relations among invariants. By converting polynomials into coefficient vectors and applying Mapper and Ball Mapper, it uncovers stable, interpretable graph patterns and connects them to classical knot-theoretic questions, such as Fox conjecture, Khovanov width, and signature versus s-invariant. It additionally examines random knots to test the generality of observed patterns and assesses how discriminative power evolves with crossing number. The work provides hypothesis-generating insights that bridge data-science visualization with rigorous knot theory, offering a scalable framework for exploring infinite knot families.

Abstract

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a way to extend mathematical exposition of various invariants of knots towards understanding their relations in statistical and cumulative way. Paper included examples illustrating how topological data analysis can illuminate structure and relations between knot invariants, state new hypothesis, and gain new insides into long standing conjectures.

Data Driven Perspectives on Knot Theory

TL;DR

The paper argues that knot invariants can be meaningfully studied as high-dimensional data, using topological data analysis tools to reveal structure and relations among invariants. By converting polynomials into coefficient vectors and applying Mapper and Ball Mapper, it uncovers stable, interpretable graph patterns and connects them to classical knot-theoretic questions, such as Fox conjecture, Khovanov width, and signature versus s-invariant. It additionally examines random knots to test the generality of observed patterns and assesses how discriminative power evolves with crossing number. The work provides hypothesis-generating insights that bridge data-science visualization with rigorous knot theory, offering a scalable framework for exploring infinite knot families.

Abstract

Data science offers a powerful tool to understand objects in multiple sciences. In this paper we utilize concept of data science, most notably topological data analysis, to extend our understanding of knot theory. This approach provides a way to extend mathematical exposition of various invariants of knots towards understanding their relations in statistical and cumulative way. Paper included examples illustrating how topological data analysis can illuminate structure and relations between knot invariants, state new hypothesis, and gain new insides into long standing conjectures.

Paper Structure

This paper contains 11 sections, 1 theorem, 2 equations, 17 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Data and methods
  3. Knot invariants as data
  4. Random knot data
  5. Mapper-like algorithms
  6. Statistical insights into the discriminative power of polynomial knot invariants
  7. Fox conjecture: illustrated and extended
  8. Ball Mapper applications in knot theory
  9. Ball Mapper vs Persistent PCA
  10. Khovanov homology, s-invariant, and signature
  11. Knot data from random polygons

Key Result

Theorem 7.1

For all simple knots (all the roots $\alpha\in\Delta(K)$ of the Alexander polynomial where $|\alpha|=1$ have multiplicity 1) up to 8 crossings and for all torus knots, the colored Jones polynomial determines the signature of the knot.

Figures (17)

  • Figure 1: Illustration of a discriminative power of the Alexander and Jones polynomials of knots up to 17 crossings where x-axis represent the multiplicity and y-axis the count with Alexander (blue) and Jones (orange).
  • Figure 2: Graphs showing the percentage of unique (left) and distinct (right) values of a single or a pair of polynomial invariants across the filtration by the crossing number for all knots (top), just non-alternating (middle) or just alternating knots (bottom row).
  • Figure 3: (A) Ball Mapper of the Alexander polynomial for all all non-alternating knots up to 17 crossings colored by the fraction of knots for which the Fox conjecture holds. (B) The distribution of the determinant values for the non-alternating knots up to 17 crossings on which the Fox conjecture holds (blue) and does not hold (orange).
  • Figure 4: Jones space of knots of up to 17 crossings colored by the fraction of alternating knots in each cluster (A), and by the crossing number (B).
  • Figure 5: Ball Mapper graph for the Jones polynomial data of knots up to $17$ crossing colored by the fraction of knots in each cluster with $16$ crossings.
  • ...and 12 more figures

Theorems & Definitions (4)

  • Conjecture 4.1: Fox trapezoidal conjecture fox1962some
  • Theorem 7.1: Garoufalidis '03
  • Conjecture 7.2
  • Conjecture 7.3