Data deposit · Dataset · version 2 of 2026-10-22, first deposited 2026-10-03

Boninger's Conjectures 1.1, 1.6 and 1.7 hold for all prime alternating knots with at most 19 crossings: a computer check

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

Permanent archive (all files, all versions): doi:10.5281/zenodo.23118923 (Zenodo, CERN).

Abstract

J. Boninger, 'An Alexander polynomial refinement for alternating links, with trapezoidal properties' (arXiv:2608.28484v1, 28 August 2026), states Conjectures 1.1 (symmetry P(x,y,z,w) = P(y,x,w,z)), 1.6 (M-convex support) and 1.7 (log-concavity) and reports them verified for all alternating knots with at most eleven crossings. We checked all three on every alternating knot in B. A. Burton's census of prime knots up to 19 crossings: 51 280 976 knots (51 280 967 hyperbolic and the nine torus knots T(2,n), n = 3, ..., 19), of which 51 280 413 have 12 to 19 crossings. No counterexample was found. The deposit contains the implementation of Boninger's polynomial P_K (direct definition, a dual-tree version and a C kernel), the per-block results and per-crossing-number summaries, and the logs of the checks: the printed examples P(4_1) and P(8_10), the paper's Theorem 4.2 and Propositions 4.3 and 4.4, agreement of the three implementations, self-tests of the conjecture checkers, and countercases for each check. One census entry per knot type suffices because mirroring and reversal act linearly on the exponents (Boninger's properties (iv), (v)) and preserve all three conjectures; this was measured, not only argued. One convention (the left/right orientation of the fundamental cycle around the basepoint) is fixed in the paper only by its figures. Our computed P agrees with Boninger's up to exchanging z and w, which leaves all three conjectures unchanged; in our implementation the setting that reproduces property (iv) is not the same for a knot and its mirror image, i.e. our orientation of the fundamental cycle is tied to the chirality of the diagram; a comparison with the author's code would settle it. SCOPE: knots only (the conjectures are stated for links); prime alternating knots only (composite knots are not in the census). STATUS: a finite computer check, not a proof; not refereed. Werner Alois Stanggassinger (sculptor) in collaboration with Claude Code (Anthropic). The census is B. A. Burton's (SoCG 2020) and is not redistributed. VERSION 2 (17 October 2026): the nine torus knots T(2,n) were added; convention sentence sharpened; author name uniform; an unrelated reference was removed from the metadata.

Keywords: knot theory · alternating knots · Alexander polynomial · Boninger polynomial · M-convexity · log-concavity · Fox trapezoidal conjecture · knot census · computer verification

Cite

@misc{stanggassinger2026boninger,
  author    = {Stanggassinger, Werner Alois},
  title     = {Boninger's Conjectures 1.1, 1.6 and 1.7 hold for all prime alternating knots with at most 19 crossings: a computer check},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23118923},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/boninger-conjectures-19-crossings/}
}

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