Code, data and Erratum for Machine learning methods and unknotting numbers

We collected the code, searches, verification files and figures relevant for the paper Machine learning methods and unknotting numbers on this page. This is joint work with Anne Dranowski, Zhen Guo and Yura Kabkov.

The paper itself is maintained separately and will be linked here once it is available. The repository contains all computational files promised in the paper: the exact lower-bound certificates, the complete minimal-diagram check for 11a14, the exploratory hard-unknot searches, and the files used for the research levels in Unknot!.

An Erratum for the paper can be found at the bottom of this page and in ERRATUM.md.

Contact

If you find any errors in the paper, please email me:

daniel.tubbenhauer@sydney.edu.au

Same goes for errors related to this page. Also, let us know if there are any questions about the code or the data.

Files in this repository

The files are divided into exact lower-bound certificates, the 11a14 verification, exploratory searches, and input data. The individual directories contain short README files with more details and commands.

Lower bounds from the Owens obstruction

The folder lower_bounds/owens_u2/ contains the signature-four calculation. It gives 245 certificates ruling out unknotting number two: 237 exact values u=3 and eight improvements from [2,4] to [3,4].

The folder lower_bounds/owens_signature_sharp/ contains the higher-signature version of the same calculation. It gives

1. 164 signature-six certificates, of which 152 prove u=4 and twelve improve [3,5] to [4,5];
2. 24 signature-eight certificates proving u=5.

All these calculations use exact arithmetic. The CSV files are the machine-readable outputs; formatted XLSX copies are included for convenience.

Montesinos correction-term obstructions

The folder lower_bounds/montesinos_u1/ contains the complete scan of the 50 parseable three-tangle Montesinos knots considered in the paper. The correction-term test obstructs unknotting number one for 49 of them. The remaining knot is 12n309, whose determinant is one.

The subfolder full_certificates/ contains one JSON certificate for each target, including the plumbing matrix, all correction terms, all compatible affine maps, and the exact reason each map fails.

The folder lower_bounds/montesinos_signature_sharp/ contains the seven higher-signature Montesinos certificates: five proving u=3 and two proving u=4.

The all-minimal-diagram check for 11a14

The folder verification/11a14/ contains the complete verification used in the paper.

The code enumerates the 17 normalized minimal diagrams of 11a14 and checks every single crossing change on every diagram. Thus there are 17 x 11 = 187 rows. The outputs record the four invariant classes that occur and resolve all apparent collisions with knots of unknotting number one using the Jones polynomial, the Alexander polynomial and hyperbolic volume.

The earlier direct two-swap experiment is also retained, but it is kept separate from the certificate used in the proof.

Hard-unknot searches

The folder searches/hard_unknots/ contains the cleaned scripts and final summaries from the larger exploratory searches. These include one-crossing-change searches in slices of the public hard-unknot data and the extended search around the 42-crossing example used in the paper.

A word of caution: these are searches for witnesses. Finding a short route gives an upper bound, but failing to find one is not a lower-bound proof. We include the negative searches for transparency and reproducibility, not as certificates.

Unknot! research levels

The folder searches/game_levels/ contains PD codes and reproducible PLink figures for the eight ten-crossing research levels

10a6, 10a11, 10a51, 10a54, 10a61, 10a76, 10a77, 10a79

and for the connected-sum challenge 4_1#9_10.

The first three playable research levels are Tortoise (10a54), Scream (10a61) and Jellyfish (10a76).

Input data and figures

The folder data/ records the KnotInfo snapshot used in the computations, the compact invariant indexes needed for the 11a14 check, and the correction to the 12n873 input discussed in the paper. The complete third-party KnotInfo table is not redistributed, but its version and checksum are recorded and the relevant indexes can be rebuilt deterministically.

The folder figures/key_knots/ contains the PLink sources and PNG files for the main knot diagrams appearing in the paper.

Running the code

The common Python dependencies can be installed with

python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install -r requirements.txt

Most scripts use the database-knotinfo package. The diagram and search code also uses SnapPy and Spherogram.

To check that all theorem-critical files are present with the expected sizes, run

python3 verify_repository.py

To check the files against the recorded checksums, run

sha256sum -c SHA256SUMS

The same structural check is run automatically by GitHub Actions. The precise commands for rebuilding individual tables are given in the README files in the corresponding directories.

A compact map from the claims in the paper to the relevant files is available in STATUS.md.

Citation

@misc{DranowskiGuoKabkovTubbenhauerUnknotData,
  author = {Dranowski, Anne and Guo, Zhen and Kabkov, Yura and Tubbenhauer, Daniel},
  title  = {Code, data and more for ``Machine learning methods and unknotting numbers''},
  year   = {2026},
  url    = {https://github.com/dtubbenhauer/unknot}
}

A machine-readable citation is provided in CITATION.cff.

Erratum

No errors in the paper are known at present.

The repository records one correction to the input data rather than to the paper: KnotInfo lists 12n873 with interval [1,3] and algebraic unknotting number two, so the interval used in the paper is [2,3]. See data/12n873_correction.csv.