Data deposit · Dataset · version 2 of 2026-10-22, first deposited 2026-07-15

Priority deposit: c7 inequality at 14/15 crossings, Knot 15331 bounds, and a blind Farey/precessing-ellipse knot atlas

Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)

Permanent archive (all files, all versions): doi:10.5281/zenodo.21389680 (Zenodo, CERN).

Note added 2026-10-17: “c7” in this deposit follows the numbering of arXiv:2605.22652v3 (May 2026), where (c7) is spP_a ≤ spF_a. In the current version v5 (September 2026) the label (c7) denotes a different inequality, and spP_a ≤ spF_a is not among the candidates (c1)–(c9).

Abstract

Three dated computational results, deposited to fix priority in permanent, non-editable, citable form. (1) The inequality spP_a <= spF_a (HOMFLY a-spread <= Kauffman a-spread; candidate (c11)/(c10)/(c7) of versions 1/2/3-4 of M. Jablonowski, arXiv:2605.22652, proved there for alternating knots and screened up to 13 crossings; version 5 no longer lists it) verified beyond 13 crossings for the first time. VERSION 2 (19 October 2026): the July tables (46,972 knots at c=14; 253,281 of the 253,293 knots at c=15) were computed without a validity check on the Kauffman polynomial, and the tool used (knotpy 0.0.1) can silently return a non-polynomial; two of their rows (14n18645 and 15nh_054728) carry a wrong Kauffman a-spread (9 instead of 10). All 195,466 non-alternating prime knots with 14 and 15 crossings in Burton's census were therefore recomputed on 17-18 October 2026 with a validity check on every row, an independent descending-skein Kauffman implementation for the 2,488 rows the tool failed on, and a second HOMFLY computation with Regina on every row: 0 violations, smallest slack 2 at c=14 and 4 at c=15, every other July row confirmed, the 12 knots absent in July included. The inequality therefore holds for every prime knot with 14 and 15 crossings (alternating ones by Proposition 3.14 of version 4). The July tables are kept byte-identical; the new table, the two corrected rows, the scripts, counter-cases, calibration and logs are added. (2) Sharper bounds and a primality/hyperbolicity argument for Knot 15331 of D. H. Fremlin's knot problem (credit already acknowledged by Fremlin, 2026). (3) A blind verification and atlas of A. Hizume's precessing-ellipse/Farey construction: 28 space curves closed at Delta=q/p (3/2..13/6), each identified blind as T(p,q) with determinant, Alexander, Jones and standard braid. HONESTY NOTE: no invariant here is computed by an AI; all numbers come from established open tools (SnapPy, SageMath, Regina, KnotJob, and an independent Kauffman implementation) and are reproducible there. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the geometry, questions and hardware are Stanggassinger's; the pipeline driving the tools and the curve-generating machines are the joint contribution. See README.md and SHA256SUMS in the archive.

Keywords: knot theory · Kauffman polynomial · HOMFLY polynomial · a-spread · torus knots · Farey sequence · Jones polynomial · hyperbolic knots · computational topology · priority deposit

Cite

@misc{stanggassinger2026homfly,
  author    = {Stanggassinger, Werner Alois},
  title     = {Priority deposit: c7 inequality at 14/15 crossings, Knot 15331 bounds, and a blind Farey/precessing-ellipse knot atlas},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.21389680},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/homfly-kauffman-spread-inequality-14-15-crossings/}
}

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