Paper · Preprint · deposited 2026-10-22
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.23570037 (Zenodo, CERN).
Ishikawa, Morifuji and Suzuki defined the twisted Alexander vanishing (TAV) order O(K) of a knot, showed O(K) >= 24, with equality exactly when the symmetric group S4 is a TAV group of K; among the prime knots with at most ten crossings, O(K) = 24 holds exactly for 9_35 and 9_46, as shown by Morifuji and Suzuki. In the sequel (arXiv:2510.25574v1, p. 3) they ask how O(K) behaves on special classes of knots, among them knots with a given crossing number. We decide O(K) = 24 for each of the 352,152,252 prime knots with at most 19 crossings in Burton's census: it holds for exactly 18,436 of them. By crossing number: 2 with 9 crossings, none with 10 or 11, 4 with 12 (12a_270, 12n_333, 12n_809, 12n_868), 5 with 13, 20 with 14, 76 with 15, 274 with 16, 789 with 17, 3,161 with 18 and 14,105 with 19; 990 of them are alternating hyperbolic, 17,396 non-alternating hyperbolic and 50 satellites, none is a torus knot. METHOD: the regular representation of S4 reduces to the four-dimensional permutation representation; for every knot all epimorphisms of the knot group onto S4 (meridians to transpositions or 4-cycles, one per conjugacy class) are enumerated on a GPU and the Wada numerator is tested at two pseudo-random points modulo 2^31 - 1; every vanishing case is certified by exact integer evaluation at 4(c-1) + 1 points, modulo enough primes to exceed twice the Hadamard bound. CHECKS: an independent CPU reference program reproduces exactly {9_35, 9_46} among the 249 prime knots with at most ten crossings; for every one of the 18,436 knots one vanishing epimorphism was re-certified on the CPU without GPU arithmetic (SageMath, 18,436 of 18,436 passed); 30 hits and 30 non-hits drawn at random were confirmed by a second script; the same CPU route decided each of the 313,230 census knots with at most 15 crossings from scratch, without using the GPU result in the decision, and found the same 107 knots, knot by knot. The result is computer-assisted: each listed knot is certified by exact arithmetic, while the absence of further knots rests on the completeness of the GPU enumeration, its modular arithmetic, the conversion of the census diagrams and Burton's census; up to 15 crossings the absence of further knots is also confirmed by the CPU program, whose enumeration and arithmetic code is independent of the GPU program; the two share Burton's census, the Wirtinger presentations and their convention, the list of knots (read from the GPU output; its length, 313,230, is the number of census knots with at most 15 crossings), the reduction to the permutation representation and the degree bound N. Composite knots were not considered. CONTENTS: the note (PDF); the list of the 18,436 knots, each with one exactly vanishing epimorphism onto S4; the per-file and per-class counts; the raw GPU output (1,805 blocks); the CPU certificates; the CPU decision of every knot with at most 15 crossings; the logs; and the programs. data/README_DATA.md describes every file with counts measured on the files. HONESTY NOTES: crossing numbers are those of Burton's census; no invariant is computed by an AI, the computations being done by Python programs using CuPy, Regina, SnapPy 3.3.2 (with knot_floer_homology 1.2.2) and SageMath 10.9. This work was developed in collaboration with the AI system Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed by Claude Code (Anthropic) working with the author. The collaboration is stated as a fact, not a disclaimer; details are in the section on tool use at the end of the note.
Keywords: knot theory · twisted Alexander polynomial · twisted Alexander vanishing order · symmetric group · knot tables · computer census · GPU computation
@misc{stanggassinger2026prime,
author = {Stanggassinger, Werner Alois},
title = {Prime knots with twisted Alexander vanishing order 24 up to 19 crossings},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.23570037},
url = {https://knot-structures.stainlesssteel4u.de/preprints/prime-knots-twisted-alexander-vanishing-order-24/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.