Data deposit · Dataset · version 3 of 2026-10-22, first deposited 2026-09-26

The Andersen–Kashaev volume conjecture for 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong

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

Permanent archive (all files, all versions): doi:10.5281/zenodo.22975395 (Zenodo, CERN).

Abstract

For 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong we exhibit an ordered ideal triangulation of the knot complement that is FAMED and geometric; by Ben Aribi-Wong, Corollary 1.9 (arXiv:2410.10776v2: 'Conjecture 1.5 is true for all FAMED geometric triangulations'), the Andersen-Kashaev volume conjecture holds for these knots. 216 distinct complements (4 chiral mirror pairs). THE THEOREM AND THE METHOD ARE NOT OURS: the conjecture is Andersen-Kashaev's (arXiv:1109.6295), FAMED triangulations and Corollary 1.9 are Ben Aribi-Wong's, the census-wide computer proof and the search code (A. Guilloux, FAMEDexploration) are Ben Aribi-Guilloux-Wong's (arXiv:2512.17437, Theorem 3.1: all but 6 of the 42 123 hyperbolic knot complements of the HT census with at most 14 crossings and 23 tetrahedra). What is deposited are computed objects and certificates. WHY NEW CASES: none of the knots lies in that census - 160 have a proven crossing number c >= 20 (so they lie outside the whole Hoste-Thistlethwaite census, up to 16 crossings), 60 were checked with SnapPy's census identification on five geometric triangulations each (calibrated on 300 census knots) against SnapPy's table of HT knots with at most 14 crossings: no match; whether any of these is an HT knot with 15 or 16 crossings was not checked. Our literature search to 25 September 2026 found no case beyond that census; not finding is not proof of absence. Triangulations have 8 to 56 tetrahedra, 106 with more than 23. THE CHAIN, PER KNOT: (1) FAMED - the authors' code only searched; the verdict is our independent implementation of Definition 1.1 (1)-(4), exact over Q, on exactly the deposited ordered triangulation; it rejects 15 of 15 deliberate perturbations. (2) Geometric - SnapPy verify_hyperbolicity (interval arithmetic) on that same triangulation. (3) Meridian - n1 of Lemma 4.1 computed with the meridian of the knot's own diagram: n1 = 0 for all 220; calibrated on five twist knots (Proposition 5.1) and on a deliberately shifted meridian. (4) Duplicates - overlapping verified volumes tested for isometric complements; the 4 isometric pairs are orientation-reversing with opposite Chern-Simons invariants, decided on complements built from the diagrams. KNOTS: 67 twisted torus knots T(p,q;2,s) (proven crossing number lower bounds 26 to 150), 10 odd pretzel knots, and knots from the constructive families of a sculptor's project; 6 sculpture records, 5 distinct knots, are among them (cosmic-love, nitro, ping-pong, spiral, sundown, trainstation; sundown and trainstation are one knot), of which only trainstation exists in stainless steel. CAVEAT: Liu-Ming-Sun-Wu-Yang (arXiv:2608.16560) prove a volume conjecture for a related invariant for every geometric triangulation; if the announced bridge to Andersen-Kashaev is built, case lists like this one may be superseded. FILES: ak_data.json (per knot: ordered triangulation as Regina tight encoding, FAMED verdict, geometric certificate, verified volume interval, meridian data, PD code), reproduce.py (coherence without dependencies; --full recomputes everything under SageMath with SnapPy and Regina, including negative controls; checker code copied verbatim), SHA256SUMS. HONESTY NOTE: no invariant is computed by an AI; every number comes from SnapPy 3.3.2, Regina 7.4, SageMath 10.9 and the FAMED checker, which is our code (written with Claude Code), included and tested against perturbations. VERSION 3 (17 October 2026): title and census statement corrected after an outside reading; data unchanged. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the questions, decisions and hardware are Stanggassinger's; the pipeline driving the tools is the joint contribution. Project: https://knot-structures.stainlesssteel4u.de/ (report, section 4, C5).

Keywords: knot theory · Andersen-Kashaev volume conjecture · Teichmüller TQFT · FAMED triangulation · ideal triangulation · hyperbolic knot · volume conjecture · twisted torus knot · SnapPy · Regina · computational topology · priority deposit

Cite

@misc{stanggassinger2026andersen,
  author    = {Stanggassinger, Werner Alois},
  title     = {The Andersen–Kashaev volume conjecture for 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.22975395},
  url       = {https://knot-structures.stainlesssteel4u.de/preprints/andersen-kashaev-volume-conjecture-220-knots/}
}

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