Data deposit · Dataset · version 2 of 2026-10-22, first deposited 2026-07-26
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.21596227 (Zenodo, CERN).
VERSION 2 (17 October 2026). An answer to the modulo-8 part of the first half of Question 9.2 of J. Conant, 'A proof of the mod 4 Kawauchi conjecture', arXiv:2607.18655 (21 July 2026), which asks which congruences modulo 8 and higher powers of 2 on the Conway polynomial follow from the presentation eps_l [l] Nabla_K = M^2 - 4 E^2 of his Corollary 7.3, beyond the constant-term constraint E(0) = (l^2-1)/8 mod 2. Writing w = z^2 and N = eps_l [l] Nabla_K, the substitution U = M - 2E, V = M + 2E turns the presentation into a factorisation N = U V with V congruent to U modulo 4 and U(0) odd, hence N = U^2 + 4 U E; modulo 2^s the class of N depends only on U modulo 2^(s-1) and E modulo 2^(s-2). With A the reduction of M modulo 2, whose coefficients are read off N by a_i = N_{2i} mod 2, three congruences are obtained: (C1) every coefficient of N in odd w-degree is even; (C2) N is congruent to A^2 modulo 4, so N mod 4 is completely determined by N mod 2; (C3) X = (N - A^2)/4 lies over F_2 in A F_2[w] + F_2[w^2], a condition on the odd w-degrees only, decided by one Gaussian elimination over F_2. NEW IN VERSION 2: together with N(0) odd, which holds for every knot, C1, C2 and C3 are NECESSARY AND SUFFICIENT for N to be representable modulo 8 by the identity of Corollary 7.3: if X = A C + D^2, then M = A + 2C and E = C + D give M^2 - 4E^2 = N mod 8 with M(0) odd. Version 1 wrongly stated that the three conditions are strictly weaker than representability. In the explicitly defined degree frames L = 1..6 (classes N mod 8 with deg_w N <= L and N(0) odd), the classes satisfying C1, C2, C3 are exactly the representable ones: 4, 16, 32, 112, 224, 896; C3 is not implied by C1 and C2 from L = 4 on (it removes 16, 32 and 128 classes at L = 4, 5, 6). The counts of version 1 (13984 classes; 4/32, 12/64, 40/256, 88/512) are withdrawn: the frame was not defined, the first number counts with multiplicity, the second counted only classes with N(0) even, and the version-1 implementation of C3 truncated a product and accepted classes that fail C3; a shipped script reproduces and decomposes them. EMPIRICAL TEST: C1, C2, C3 hold, and hence representability modulo 8, on 315 of 315 knots of version 3 of the companion deposit 10.5281/zenodo.21934239 (rotary-reflection order 4: 18/18, order 8: 261/261, order 16: 36/36); the original test of version 1, 14 of 14 knots of 10.5281/zenodo.21590761 with l in {-7, -3, 1, 5, 9}, stands, and these fourteen are among the 315. The corpus table of version 1 (547 Conway polynomials) is withdrawn because the script behind it was not deposited and could not be recovered. OPEN: whether every class satisfying C1-C3 is realised by an actual knot in the setting of Corollary 7.3; the higher powers of 2; the second half of Question 9.2, whether E is a well-defined invariant with a topological interpretation, which depends on Conant's construction of E from Q_1 rather than on the identity; Theorem 7.1 is not treated. NO NEW THEOREM ABOUT KNOTS IS CLAIMED; this is an elementary but apparently unrecorded consequence of Conant's identity, deposited with data and code. reproduce.py (Python 3 standard library only, about 6 seconds) re-derives the claims of the note; q92_frame_check.py covers frame 6 and c123_on_315.py the 315 knots; all outputs are shipped. HONESTY NOTES: crossing numbers quoted are upper bounds; no invariant is computed by an AI, the knot-theoretic inputs coming from Regina, SnapPy/spherogram, SageMath and SymPy in the companion deposits, and everything here being exact integer arithmetic that can be read in full. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic): the sculptures, geometries, questions and hardware are Stanggassinger's; the pipeline, the corpus and the derivation are the joint contribution. See README.md, NOTE.md (section 'Changes in version 2') and verification_log.md, which records the errors of version 1 and how they were found.
Keywords: knot theory · Conway polynomial · amphicheiral knots · Kawauchi conjecture · congruences modulo powers of two · Conant Corollary 7.3 · quantum integer · computational topology · priority deposit
@misc{stanggassinger2026conway,
author = {Stanggassinger, Werner Alois},
title = {Congruences on the Conway polynomial forced by Conant's Corollary 7.3: an answer to the first half of Question 9.2},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.21596227},
url = {https://knot-structures.stainlesssteel4u.de/preprints/conway-polynomial-congruences-conant/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.