Data deposit · Dataset · deposited 2026-10-26
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.23660288 (Zenodo, CERN).
Accepted into the OEIS on 2026-10-29: the terms a(9) = 873,994,338 and a(10) = 109,051,496,360 computed here are now part of A163566 (approved by an OEIS editor), credited to Werner Alois Stanggassinger with a link to this deposit.
Two new terms of OEIS A163566, "Total number of grid diagrams of nontrivial knots on grid size n" (Baldridge and McCarty, arXiv:0907.5401v1): a(9) = 873,994,338 and a(10) = 109,051,496,360, both computed exactly. The OEIS entry (revision 8) has the four terms n = 5..8; our code reproduces all four. METHOD: every grid diagram of size n (pair of permutations X, O with X(i) != O(i) whose induced permutation is one n-cycle) is enumerated on one GPU (CuPy, RTX 3070). The Alexander polynomial is tested through the minesweeper determinant det M(G) = +-t^k (1-t)^(n-1) Delta(t) (Manolescu, Ozsvath, Szabo, Thurston, arXiv:math/0610559v3, Proposition 6.2), first mod p = 2^31-1 at three points, then exactly for every grid that test leaves as trivial (interpolation on a per-grid exponent window, coefficients bounded by n! < p/2, overflow counter 0). At n = 9 and n = 10 the exact pass found 0 nontrivial grids among those flagged trivial. WHY DELTA = 1 MEANS UNKNOT HERE: no nontrivial knot with Delta = 1 has arc index <= 10 -- prime knots by KnotInfo (crossing number <= 13) and the 48 knots of Jin-Park arXiv:1010.2916v1 Table 1 (crossing number >= 14, DT codes decoded with KnotScape 1.01), composite knots by Cromwell's formula alpha(K1#K2) = alpha(K1) + alpha(K2) - 2 (Topology Appl. 64 (1995); statement read in arXiv:2509.21274v2, not in the original). CALIBRATION, fresh runs 24 October 2026: n = 5..8 give 10, 972, 85022, 8077072 in the mod-p run, the exact pass, and an exact-only run; a deliberately wrong target makes every flagged grid nontrivial, so the checker can fail. Run time n = 10: 6.5 h enumeration, 76.4 h exact pass. NO NEW THEOREM IS CLAIMED. Every number comes from the scripts included and established open tools (CuPy, numpy, SnapPy, KnotInfo, KnotScape); no invariant is computed by an AI. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the question and the hardware are Stanggassinger's; the code and the verification were written and run jointly. Code GPL-3.0-or-later, data and text CC BY 4.0. See README.md and SHA256SUMS in the archive.
Keywords: knot theory · grid diagrams · Alexander polynomial · arc index · OEIS · GPU enumeration
@misc{stanggassinger2026grid,
author = {Stanggassinger, Werner Alois},
title = {Grid diagrams of nontrivial knots: a(9) and a(10) of OEIS A163566},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.23660288},
url = {https://knot-structures.stainlesssteel4u.de/preprints/grid-diagrams-nontrivial-knots-9-10-oeis/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.