Data deposit · Dataset · version 2 of 2026-10-22, first deposited 2026-09-22
Werner Alois Stanggassinger — in collaboration with Claude Code (Anthropic)
Permanent archive (all files, all versions): doi:10.5281/zenodo.22893039 (Zenodo, CERN).
Version 2 (17 October 2026). One symmetric ribbon number determined by an explicit witness, r_s(12n706) = 4; a candidate upper bound for 12n504 that the source's Lemma 2.3 does not cover as written; and a computed subset of the set R^s_5. NO THEOREM OF OURS IS CLAIMED AND NONE OF THE MATHEMATICS IS OURS: the notion of a symmetric ribbon number, the lemmas used, the bounds and the tables belong to Akash, Corrado, Placke, Sanketh, Starns, Timothy and Zupan, 'Symmetric ribbon numbers of low-complexity knots', arXiv:2606.02390v1 (1 June 2026; still v1 when checked on 17 October 2026). What is deposited is a set of computed objects together with the checks they survived and an explicit statement of what they do and do not establish. WHAT CHANGED IN VERSION 2. Version 1 claimed two settled cases, 12n706 and 12n504. A checker written independently of the code that built the witnesses (17 October 2026, included in this version) confirms 12n706 and does not confirm 12n504 as stated: in both 12n504 witnesses one of the five ribbon singularities violates the 'globally consistent' proviso in the proof of Lemma 2.3, and making it consistent changes the knot. Version 1 also called the deposited DT codes symmetric unions, which they are not, and said that the two results did not rest on the calibration of our constructor, which was wrong: an upper bound needs a valid labeled knotoid diagram, not only a verified boundary. The data file is unchanged; the README lists every change. RESULTS. 12n706: r_s = 4. The lower bound is the source's (Table 3 gives r = 4, Lemma 2.2 gives r_s >= r; independently Corollary 5.1 excludes r_s = 2 and 3 for det = 49). The upper bound is a labeled knotoid diagram with four crossings whose band disk has four ribbon singularities, all globally consistent, so Lemma 2.3 applies as written. Alexander Zupan, an author of the source, confirmed r_s(12n706) = 4 from our DT code (personal communication, 23 September 2026). 12n504: the source gives r_s in {5, 6}; r_s = 5 is a candidate. Our labeled knotoid diagram with five crossings gives a band disk whose boundary is 12n504, but one of its five singularities is not globally consistent, so r_s(12n504) <= 5 does not follow from Lemma 2.3 as written. It would follow if the two singularity choices at a crossing are the same modular piece up to a half turn in the plane; 120 test constructions support this; it has not been checked by a knot theorist, and Zupan has not confirmed 12n504. WHAT THE WITNESSES ARE. Each is given in full: Gauss word of the arc diagram, twist labels, and the DT code of the boundary of the band disk built from the knotoid (4c + sum of |labels| crossings). The DT code is not a symmetric union diagram with c off-axis crossing pairs; by Lamm's theorem (det K = det(J)^2) no such diagram exists for det 49 or 121. The bound is r_s <= number of ribbon singularities = number of crossings of the knotoid. The signs of the twist labels follow an edge convention, not the handedness of the twists. HOW IDENTITY IS ESTABLISHED: not by matching polynomials. For every witness deposited, SnapPy's isometry_signature(verified=True) agrees bit-for-bit with that of the named knot - an interval-arithmetic certificate - and verify_hyperbolicity() returns True on both sides; for 12n706 the verified volume is 13.4173743919? on both sides. The isometry signature does not see mirror images: one 12n504 witness bounds the mirror image, which leaves r_s unchanged. R^s_5. The source states on page 3: 'If r_s(K) = 5, the set R^s_5 is too large to compute via our methods.' We computed 145 Alexander polynomials, deposited as coefficient lists (R^s_4 has 27); determinants occurring: 1, 9, 25, 49, 81, 121, 169. This is a computed subset of R^s_5, complete if our enumeration of the 26 five-crossing arc diagrams is complete, which is not calibrated. Two checks make completeness plausible, not proven: Theorem 1.6 of the source proves by entirely different means (Kidwell's theorem on the z-degree of the Kauffman polynomial) that r_s(K) = 5 implies det(K) <= 169, and the largest determinant in the computed set is exactly 169; and every knot in Table 3 with r_s = 5 proven has its Alexander polynomial in the set, 17 of 17. Membership rests on our constructor, calibrated by exact reproduction of the source's R^s_2 (2), R^s_3 (6) and R^s_4 (27 of 27) plus 12/12 negative and 5/5 positive controls; its rule for choosing between the two singularities at a crossing is not the source's consistency rule, and the Alexander-based calibration cannot tell the two rules apart. NOT CLAIMED: r_s(12n504) = 5; any exclusion of the form 'Delta not in R^s_5 implies r_s != 5', in particular r_s(12a1202) = 6, which remains a reasoned candidate; and Delta in R^s_n does NOT imply r_s = n, since Delta is not a complete invariant. The source uses these sets as an obstruction only, and so does this deposit. STILL OPEN: 12n504 (candidate above). 12n49: det 81, so Corollary 5.1 gives r_s >= 5 (Table 3 of the source gives 5 to 8); no witness with five crossings found for twist labels |a| <= 5; r_s(12n49) remains open. 12n23 is decidable with the same machinery and is not decided (the run died without a traceback inside SnapPea's C code). Ten further knots were searched for witnesses with five crossings at |a| <= 5 without success; absence of a witness in a bounded search is not an exclusion. FILES: rs_data.json (byte-identical to version 1; the README notes which fields are misleading); a dependency-free reproduce.py that re-checks the internal coherence of the data file and prints what it cannot verify without SnapPy; and the independent checker (independent_checker.py with controls.py and three driver scripts) with its original output: a search for symmetric union symmetries (none found in any deposited diagram), a recognition of band disks with ribbon singularities, the global-consistency test, and the counter-cases and controls it was tested on. It needs SageMath with SnapPy and spherogram. HONESTY NOTE: no invariant in this deposit is computed by an AI. Every number comes from established open tools - SnapPy/spherogram and SageMath - and is recomputable from the diagrams included here; the independent checker is our own code, tested on the controls it ships with. This work was done by Werner Alois Stanggassinger in collaboration with Claude Code (Anthropic), which orchestrates the computation: the questions, the decisions and the hardware are Stanggassinger's; the pipeline driving the tools is the joint contribution. The knot 12n706 is the knot of one of Stanggassinger's sculptures, Celtic-for-Playing (a rendered design, not a steel piece); that is how the case was chosen and it plays no role in the mathematics. Prior art was checked after the computation, not before, on 19 September 2026, and the version status of the source again on 22 September and 17 October 2026.
Keywords: knot theory · symmetric ribbon number · symmetric union · ribbon knot · labeled knotoid diagram · Alexander polynomial · isometry signature · SnapPy · computational topology · hyperbolic geometry · priority deposit
@misc{stanggassinger2026symmetric,
author = {Stanggassinger, Werner Alois},
title = {Symmetric ribbon numbers: r_s(12n706) = 4, a candidate for 12n504, and a computed subset of R^s_5},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.22893039},
url = {https://knot-structures.stainlesssteel4u.de/preprints/symmetric-ribbon-numbers-12n706-12n504/}
}This page mirrors the Zenodo record; if the two ever differ, the Zenodo record is authoritative.