Paper · Preprint · version 2 of 2026-11-03, first deposited 2026-10-22

Two of Jabłonowski's candidate inequalities hold for all prime knots with at most 18 crossings

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

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Permanent archive (all files, all versions): doi:10.5281/zenodo.23566264 (Zenodo, CERN).

Abstract

Jabłonowski (arXiv:2605.22652) proposed nine candidate inequalities between integer knot invariants and proved structural special cases. Two of them compare the concordance invariants of Ozsváth–Szabó and Rasmussen with the z-degree of the HOMFLY polynomial: 2|tau(K)| <= deg_z P(K) and |s(K)| <= deg_z P(K). We show by an exact computation that both hold for every prime knot with at most 18 crossings, and hence for every connected sum of such knots. The genus and tau come from knot Floer homology (Szabó's program in SnapPy), s (where it is needed) from Schütz's KnotJob, and deg_z P from Regina, cross-checked with SageMath wherever it decides the outcome. By Jabłonowski's results and the argument of his Proposition 5.19, only the non-alternating knots with deg_z P < 2g need tau and s; there are none up to 14 crossings and 34,974 with 15 to 18 crossings. On 2,331 of them the inequalities are sharp; at least 2,254 of these lie outside both proof mechanisms of his paper. On 152 sharp knots a short chain of genus-one moves, together with tau, determines the four-ball genus, g_4 = deg_z P/2 < g. The deposit contains the paper (PDF and LaTeX source), the per-knot data of all 34,974 genus-gap knots (diagrams, g, deg_z P, tau, s, sigma, a second diagram and, for the 16,498 knots with 18 crossings where that repeated the first, a third), the counts per crossing number, the certificates of the sharp knots, and the scripts.

Keywords: Rasmussen s invariant · HOMFLY polynomial · four-ball genus · knot tables · computer-assisted proof · knot theory · Ozsváth–Szabó tau invariant

Cite

@misc{stanggassinger2026jablonowski,
  author    = {Stanggassinger, Werner Alois},
  title     = {Two of Jabłonowski's candidate inequalities hold for all prime knots with at most 18 crossings},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23566264},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/jablonowski-inequalities-18-crossings/}
}

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