Paper · Preprint · version 4 of 2026-10-22, first deposited 2026-10-05

Vol-Det, Fox, Hoste and a conjecture of Hirasawa–Murasugi for all prime alternating knots with at most 19 crossings

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

Full text (PDF)

Permanent archive (all files, all versions): doi:10.5281/zenodo.23156040 (Zenodo, CERN).

Abstract

For every prime alternating knot with at most 19 crossings in B. A. Burton's census (51 280 976 knots, of which 51 280 967 are hyperbolic) we verify: the Vol-Det conjecture of Champanerkar, Kofman and Purcell, vol(K) < 2π log det(K) (for the hyperbolic ones), published for up to 16 crossings; Fox's trapezoidal conjecture in its strict form; log-concavity of the Alexander coefficients; Hoste's conjecture that every zero of the Alexander polynomial has real part > -1; and Conjecture 5.11 of Hirasawa and Murasugi on the width of the plateau. The verification is done knot by knot: Vol-Det by Adams' diagrammatic volume bound (double precision, with a margin far above rounding) and an exact determinant, with 72 remaining knots settled by verified interval volumes; the polynomial conditions on exactly computed Alexander polynomials, Hoste by an integer Routh–Hurwitz test. In particular, given the completeness of the census, a counterexample to Hoste's conjecture among prime alternating knots needs at least 20 crossings. The deposit contains the note (PDF and LaTeX source), the code, the summaries and logs of all runs, and per-knot files for the exceptions; the per-knot arrays over all knots (about 8 GB) are reproducible with the code. Version 4 adds, for every crossing number from 4 to 19, the knots with the largest ratio vol/2π log det and the smallest difference 2π log det − vol over all knots, with verified volumes, weaving knots and the share of quadrilateral faces; the volume of every knot is in volumes.zip.

Keywords: knot theory · alternating knots · Vol-Det conjecture · Fox's trapezoidal conjecture · Hoste's conjecture · Alexander polynomial · hyperbolic volume · knot census

Cite

@misc{stanggassinger2026alternating,
  author    = {Stanggassinger, Werner Alois},
  title     = {Vol-Det, Fox, Hoste and a conjecture of Hirasawa–Murasugi for all prime alternating knots with at most 19 crossings},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23156040},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/alternating-knots-19-crossings-vol-det-fox-hoste/}
}

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