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

arXiv: https://arxiv.org/abs/1912.10086 · HTML: https://mlsanity.com/paper/1912.10086/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2019.

## Built on by (most cited first, 7 of 7)

- [Learning knot invariants across dimensions](https://mlsanity.com/paper/2112.00016/) (2021)
- [Machine learning the dimension of a Fano variety](https://mlsanity.com/paper/2309.05473/) (2023)
- [Mapper-type algorithms for complex data and relations](https://mlsanity.com/paper/2109.00831/) (2021)
- [Big data comparison of quantum invariants](https://mlsanity.com/paper/2503.15810/) (2025)
- [On detection probabilities of link invariants](https://mlsanity.com/paper/2509.05574/) (2025)
- [On knot detection via picture recognition](https://mlsanity.com/paper/2510.06284/) (2025)
- [Colored Jones Polynomials and the Volume Conjecture](https://mlsanity.com/paper/2502.18575/) (2025)

## Abstract (arXiv)

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.