Data deposit · Dataset · version 5 of 2026-10-24, first deposited 2026-07-25
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.21590760 (Zenodo, CERN).
Version 5 (prepared 23 October 2026): corrections to the text of version 4 (compiled 19 October 2026, on Zenodo 22 October 2026); data and existing scripts are unchanged, and no number of version 4 changes. The main correction is CORRECTION IN VERSION 5 below; the others are the sign convention of l, the rotation parameter (it depends on the choice of generator), the scope of the cited results of Conant, and the honesty note; the full list is in README.md. Version 4 superseded version 3 (315 knots) on the same concept DOI; the 315 records of version 3 are carried over unchanged in every value. Here: 338 distinct hyperbolic knots that are positive amphicheiral but not strongly so -- each carries an orientation-reversing symmetry of order 2^a preserving the knot orientation, and carries no orientation-reversing involution, no negative amphicheiral symmetry and no inversion, so that neither Hartley's theorem nor Hartley-Kawauchi applies. For each, the Conway polynomial does not split as f(z)f(-z) with f in Z[z]. 26 lie at a=2 (rotary reflection of order 4), 276 at a=3 (order 8) and 36 at a=4 (order 16). NO NEW THEOREM IS CLAIMED. What is deposited is a generator for this class together with verified data. WHAT IS NEW IN VERSION 4. (1) 23 further knots: 8 at order 4, all with rotation parameter c = 3 (every earlier order-4 knot has c = 1), and 15 at order 8. They come from a generation run of 15-16 October 2026 that collected leading coefficients for Conant-Manathunga Question 4.8; each passed the same four checks as the 315 (embedding and Alexander polynomial; geometry, symmetry group and deduplication by isometry; a second, independent Alexander computation; solvability of the identity of Conant's Corollary 7.3), and each has a volume more than 1e-6 away from every one of the 315. (2) Question 4.8: the 338 realise 87 distinct leading coefficients (version 3: 66), from -988869 to +309181737. (3) Question 4.9: the prime values realised here are -29, -13, -7, -5, -3, 5, 7, 13. Table 1 of Conant-Manathunga (arXiv:1608.04453) realises -3, 5, -7, -11, 13, 17; the absolute value 29 is not among them and is realised here by one knot, SK2_3_1767_43_07 (order 4, leading coefficient -29, diagram with 44 crossings, volume 43.8993). What we checked for other realisations: Table 1 and Conant, arXiv:2607.18655v2 (one of the two works citing Conant-Manathunga in Semantic Scholar on 19 October 2026; the other, arXiv:2010.05120, we did not read); other literature was not searched. Version 3 said that it added nothing beyond Table 1, which was true for its 315. (4) The congruences C1-C3 of our deposit concept doi:10.5281/zenodo.21596227 (its version 1 tested them on 14 knots, its version 2 of 22 October 2026 on the 315 of version 3) hold on all 338 for a suitable sign (26/26 at order 4, 276/276 at order 8, 36/36 at order 16); script and output are included. They are necessary conditions for the identity of Corollary 7.3 and so follow from its solvability; their value is a check recomputable from the archive. (5) Data field names are now English; the old-to-new table is included. (6) The NOTE of version 3 gave 3^-1 mod 16 as 3; it is 11. THE ROTATION PARAMETER. Corollary 7.3 of Conant is quoted from arXiv:2607.18655v2 (v1 states it for order 4 only). Lemma 4.2 of Conant, arXiv:2607.18655v1, states c*l = 1 mod 2^a (v2 states only that l is odd). For a <= 3 every unit of Z/2^a is its own inverse, so c*l = 1 and l = c are the same condition there and our data cannot separate them. At order 16 they separate: all 36 order-16 knots have l = c, so c*l = c^2 = 9 is not 1 mod 16, at all four parameters c in {3, 5, 11, 13}; over all 338, c*l = 1 holds for 302 with our c and our signed l (47 knots have l < 0; Conant orients so that l > 0, and the sign of the rotation parameter depends on the orientation of the coordinates in the same way). That l = c is forced by the construction (the xy-modes are n = c mod 2^a, and then the winding number l of the xy-projection is = c mod 2^a as well). A rotation parameter belongs to a generator, not to the knot: every odd power sigma_c^j generates the same group, with parameter c*j (an included script illustrates this; it is implied by the construction certificate and cannot fail); for sigma_c (rotation number 1 on the knot) the congruence c*l = 1 fails at order 16, and for the generator with parameter c^-1 it holds on all 36 order-16 knots. The proof of Lemma 4.2 in v1 derives c'*l = c mod 2^a, with c' the rotation number of the generator on K and c its parameter; for our generators this gives l = c. The same proof first replaces g "by the generator g^c'' with c'c'' = 1" and closes "since c' is the inverse rotation datum one obtains cl = 1". Which generator the displayed congruence refers to is for Conant to say. Versions 3 and 4 presented a discrepancy and said the reading 'our c is his c^-1' had passed a test (both inverse pairs mod 16 realised with l = c); since l = c is forced by the construction, that was not a test. WE DO NOT CLAIM A COUNTEREXAMPLE TO LEMMA 4.2. CORRECTION IN VERSION 5: versions 1-4 reported the constant-term congruence E(0) = (l^2-1)/8 mod 2 as a check, and versions 2-4 called the branch l = +-3 mod 8 the place where it could fail (7 of 7 plus 73 of 73, 121 of 121 and 132 of 132 satisfied, respectively). It is not one. At z = 0 the identity eps*[l]*Nabla = M^2 - 4E^2 reads eps*l = M(0)^2 - 4E(0)^2; with l odd, M(0)^2 = 1 mod 8, and this forces E(0) = (l^2-1)/8 mod 2 for every integer solution, in both branches. The 338 of 338 is therefore implied by the solvability of the identity (338 of 338); a brute-force confirmation is included. This is not our observation: Conant states and proves it for order 4 in arXiv:2607.18655v1, Corollary 1.6 ("reducing constant terms modulo 8, the parity of E(0) is determined by the linking number"), and for all a >= 2 in v2, Corollary 7.3. NULL MEASUREMENT: since every curve carries sigma_c by construction, Conant's v2 Theorem 1.3 already guarantees that the identity is solvable, so 338 of 338 is a consistency check of solver and data; the null rows show the solver is not always-yes: the same solver finds a presentation for 0 of 200 random polynomials with p(0)=1 and for 11 of 144 genuinely chiral hyperbolic knots (7.6 percent), against 338 of 338 here; the two null rows were first reported in version 2, re-measured for version 3 and not repeated for version 4. CROSSING BOUNDS are, for every knot, the crossing number of the diagram included with it, 19 to 256; no minimality is claimed. Volumes 12.814 to 230.159. A GUARD checks that the diagram shipped with a knot carries that knot's own volume (Mostow): 338 of 338, with a self-test that swaps two diagrams. The construction-certificate defect is at most 7.51e-14. A per-candidate deadline used during generation does not cut at random: in the generation of the 315, among candidates it had killed and that were later completed, 225 of 1197 are counterexamples (18.8 percent), against 1602 of 41893 evaluated at first sight (3.8 percent), so deadline-limited counts understate in a biased direction. Each knot is a closed-form Fourier curve invariant under sigma_c = R_z(2 pi c / 2^a) composed with z -> -z, so positive amphicheirality is a property of the construction rather than of a computation; six numbers per knot reproduce the curve bit-exactly and a standalone numpy-only script is included, carrying a self-test that must fail against the wrong rotary reflection. HONESTY NOTE: no invariant here is computed by an AI; the knot invariants come from established open tools (Regina, SnapPy/spherogram, SageMath, SymPy) and are reproducible there from the planar diagrams included. The construction certificate, the linking number l, the C1-C3 test (both scripts included), the Corollary 7.3 solver, the second Alexander computation, the reading of symmetry types from the cusp matrices, the splitting decision and the volume guard are our own code; the generation statistics are reported, not recomputable from the archive. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the geometry, the questions and the hardware are Stanggassinger's; the pipeline driving the tools and the curve-generating machines are the joint contribution. See README.md, NOTE.md, verification_log.md and SHA256SUMS in the archive.
Keywords: knot theory · amphicheiral knots · Conway polynomial · Kawauchi conjecture · rotary reflection · Fourier knots · hyperbolic knots · leading coefficient · computational topology · priority deposit
@misc{stanggassinger2026amphicheiral,
author = {Stanggassinger, Werner Alois},
title = {Priority deposit v5: 338 positive amphicheiral hyperbolic knots whose Conway polynomial does not split -- Part III, order 16 and a discrepancy in the rotation parameter},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.21590760},
url = {https://knot-structures.stainlesssteel4u.de/preprints/amphicheiral-knots-conway-polynomial-splitting/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.