Learning bridge numbers of knots

TL;DR

This paper uses computational methods and machine learning to determine and analyze the bridge numbers of classical and virtual knots, revealing differences and creating large labeled datasets for future research.

Contribution

It introduces a comprehensive dataset of over one million labeled knots and evaluates machine learning models for classifying their bridge numbers, highlighting distinctions in virtual knots.

Findings

Machine learning models can classify bridge numbers with high accuracy.

Differences between classical and virtual knots' bridge numbers can be arbitrarily large.

Large-scale datasets enable new computational approaches in knot theory.

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.