# AI provenance: The bridge index and the nonorientable four-genus of knots with at most thirteen crossings

Author: Deep Bhattacharjee. No AI system is an author or contributor.

## How the work was produced

Deep Bhattacharjee set the task, closing open entries of the KnotInfo tables, and directed the work. Claude (Anthropic), through Claude Code:
- chose the bridge-index entries [3,4] at thirteen crossings and the open nonorientable four-genus entries at eleven to thirteen crossings;
- designed and ran the searches: transvection and reflection colourings over finite fields; pinch moves on random diagrams, identified with SnapPy; the exhaustive search for embeddings of Goeritz lattices;
- wrote the certificates and the independent verifiers;
- drafted and revised the paper, which first appeared as two papers and is now joined into one.

The paper states this in its section "Use of AI tools".

## How the results are checked

Every bound is a certificate that a program independent of the search checks:
- Bridge index: verify.py, standard library only, checks for each of the 243 knots the Wirtinger relations, the rank conditions and the Wirtinger number 4.
- Pinch moves: verify.py, built on Regina without SnapPy, checks each of the 2,728 pinch certificates by Reidemeister moves or by an isomorphism of triangulated complements. It rejects four kinds of corrupted certificate.
- Lattice lower bounds: verify_lower.py recomputes the 382 lattice lower bounds with the search order shuffled. Against the alternating knots with at most eleven crossings, the bound never exceeds the γ₄ value recorded in KnotInfo; with the sign of σ reversed it contradicts that value more than two hundred times.

## What is not claimed

- 293 bridge-index entries and 988 four-genus entries stay open. The paper lists them together with the searches that failed.
- The four-genus lower bounds cover only alternating knots.
- No general conjecture is proved. The meridional rank conjecture is confirmed only for the 243 knots.

## Labels given on the Hexagon form

- Text of the contribution: a mix of human-written and AI-generated.
- Human understanding at submission: some parts, but not all, checked and understood by a human.

## Where everything is

https://github.com/creelie/conjecture-to-choose, release v1.3.0, archived on Zenodo, doi:10.5281/zenodo.23561157. The certificates and verifiers are also in anc/ of the source ZIP.