Knot Invariants Data
Description
This repository contains a collection of datasets for various knot invariants, including polynomial and homological data. Each dataset is stored in .csv format with coefficients or values organized by columns for ease of analysis and computational use.
📊 Included Invariants
| Invariant | Description |
| Alexander polynomial | The first knot polynomial by J.W. Alexander in 1923. |
| Jones polynomial | Knot polynomial by V. Jones arising from representation theory of $\displaystyle U_{q}({\mathfrak {sl}}_{2})$. |
| Theta polynomial | Very strong, computable, and fun 2-variable polynomial by D. Bar Natan and R. Van Der Veen. |
| HOMFLYPT polynomial | Two-variable polynomial generalizing both Alexander and Jones polynomials. |
| Knot Floer homology | Categorification of the Alexander polynomial by Ozsvath-Szabo (2004) and Rasmussen (2003) based on Heegaard Floer homology. |
| Khovanov homology | A categorification of the Jones polynomial by M. Khovanov (1999). |
📁 Dataset Format
Each dataset is provided as a .csv file. Each row typically corresponds to a knot, and columns represent:
- Knot identifier
- Polynomial coefficients or homology ranks
- Additional scalar invariants (crossing number, alternating, signature, s invariant, etc.)
All datasets use the same knot identifier, following KnotScape's ordering for knots up to 16 crossings (e.g. `13n5109`) and Regina's ordering for 17 crossings knots (e.g. `17ns_29`). In general the knot identifier is of the form c[an][tsh]_k, where:
- c is the number of crossings;
- [an] indicates whether the knot is alternating or non-alternating (only for knots with more than 10 crossings);
- [tsh] indicates whether the knot is a torus, satellite or hyperbolic knot (only for 17 crossings knots);
- k is a positive integer that sorts the knots within each of these classes.
Please refer to each subdirectory’s README.md for format-specific notes.
📦 Repository Structure
/<root>├── data/│ ├── Alexander/| ├── HFK/│ ├── HOMLFYPT/│ ├── Jones/│ └── Khovanov/│ ├── Theta/├── README.md├── LICENSE└── utils/ (optional: scripts used for generating and parsing the data)
🧪 Credits and Data Generation
Alexander, Jones and HOMFLYPT data were generated using the KnotTheory package.
Khovanov homology data were generated by Dirk Schütz with KnotJob using PD-codes from Knotscape and Regina.
HFK data were generated by Davide Gurnari using Snappy from the PD-codes provided by Dirk Schütz.
Theta date were generated by Dror Bar-Natan and Roland van der Veen from a Mathematica notebook available at https://drorbn.net/AcademicPensieve/Projects/Theta. A second implementation, using Python and SageMath is available at https://www.rolandvdv.nl/Theta.
📄 License
This project is licensed under the MIT License.
Files
data.zip
Additional details
Funding
- U.S. National Science Foundation
- Collaborative Research: Computational Topology and Categorification of Cancer Genomic Data: Theory and Algorithms 1854705
- National Science Centre
- Dioscuri Program
- Max Planck Society
- Dioscuri Program