Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial

TL;DR

This paper applies manifold learning techniques to analyze the structure of the Jones polynomial across a vast dataset of knots, revealing it as an approximately 3-dimensional manifold with stable properties.

Contribution

It introduces a novel filtration-based method for analyzing infinite knot data sets and demonstrates its effectiveness in revealing the manifold structure of the Jones polynomial.

Findings

Jones polynomial forms an approximately 3D manifold

The manifold structure is stable across different filtrations

Results suggest additional structures worth exploring

Abstract

We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data we find that it can be viewed as an approximately 3 dimensional manifold, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.