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Learning bridge numbers of knots

Hanh Vo, Puttipong Pongtanapaisan, Thieu Nguyen

TL;DR

The paper investigates bridge numbers for classical and virtual knots, highlighting that $b_1$ and $b_2$ can diverge in the virtual setting. It combines combinatorial tools (Gauss codes and biquandles), algebraic invariants (meridional rank), and large-scale data generation to bound and predict bridge numbers. A key contribution is a biquandle-based lower bound for $b_2$ and a Kishino-type construction showing unbounded gaps between $b_1$ and $b_2$, complemented by extensive datasets with over a million labelled examples and a Random Forest classifier achieving high accuracy on later-stage predictions. The work also discusses an upper bound via the virtual braid index and outlines future directions for improved invariant estimation and ML-driven knot analysis with broader applicability.

Abstract

This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple natural definitions of bridge number, and we demonstrate that the difference can be arbitrarily far apart. We then acquired two datasets, one for classical and one for virtual knots, each comprising over one million labeled data points. With the data, we conduct experiments to evaluate the effectiveness of common machine learning models in classifying knots based on their bridge numbers.

Learning bridge numbers of knots

TL;DR

The paper investigates bridge numbers for classical and virtual knots, highlighting that and can diverge in the virtual setting. It combines combinatorial tools (Gauss codes and biquandles), algebraic invariants (meridional rank), and large-scale data generation to bound and predict bridge numbers. A key contribution is a biquandle-based lower bound for and a Kishino-type construction showing unbounded gaps between and , complemented by extensive datasets with over a million labelled examples and a Random Forest classifier achieving high accuracy on later-stage predictions. The work also discusses an upper bound via the virtual braid index and outlines future directions for improved invariant estimation and ML-driven knot analysis with broader applicability.

Abstract

This paper employs various computational techniques to determine the bridge numbers of both classical and virtual knots. For classical knots, there is no ambiguity of what the bridge number means. For virtual knots, there are multiple natural definitions of bridge number, and we demonstrate that the difference can be arbitrarily far apart. We then acquired two datasets, one for classical and one for virtual knots, each comprising over one million labeled data points. With the data, we conduct experiments to evaluate the effectiveness of common machine learning models in classifying knots based on their bridge numbers.
Paper Structure (14 sections, 2 theorems, 9 equations, 11 figures, 3 tables)

This paper contains 14 sections, 2 theorems, 9 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Knots and their presentations
  4. The bridge number: a geometric complexity
  5. The meridional rank: an algebraic complexity
  6. Estimation from biquandles: difference between $b_1$ and $b_2.$
  7. Bounding version 2 bridge numbers for virtual knots:
  8. Implementation
  9. Data set
  10. Classical knots
  11. Virtual knots
  12. Behaviors of the Jones polynomial under local moves
  13. Experiment with machine learning
  14. Conclusions

Key Result

Proposition 3.1

Let $X$ be a biquandle of order $n$, then $n^{b_2(K)}\geq col_{X}(K)$

Figures (11)

  • Figure 1: A diagram with Gauss code $\{-1,2,-3,1,-2,3\}$.
  • Figure 2: A nonplanar Gauss code
  • Figure 3: A diagram with Gauss code $\{-1,2,-4,4,-3,1,-2,3\}$. Note that move 1 from Definition \ref{['def:orientedgauss']} was performed.
  • Figure 4: Demonstrating the coloring method, showing that $b_1(K)=1.$
  • Figure 5: For a coloring to be legitimate, these relations have to be satisfied at every crossing.
  • ...and 6 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.1
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • proof
  • Theorem 3.1
  • ...and 1 more