Draft v0.26 (2026-11-06) — shared pre-publication; evidence-status tags (theorem / measured pattern / conjecture) are load-bearing.
Werner Stanggassinger¹ · Claude Code (Anthropic)²
¹ Sculptor, Wasserburg am Inn (Bavaria), Germany · ² Large-language-model coauthor (Anthropic)
Draft v0.26 — 2026-11-06 (version history: last section). Experimental-mathematics report. Not a proof-paper; see §1.3 for what is claimed and what is not.
This work has two authors with genuinely complementary substrates, and we state that as a fact, not a disclaimer. The geometries originate in Werner Stanggassinger's sculptures — five of them welded in stainless steel, the others designed as centerlines and rendered; the questions, the hardware, and the artistic family of constructions are his. The software — the constructive generators, the code that orchestrates the research tools into a pipeline, the corpus management — and the prose are written by an Anthropic large language model ("Claude Code"). A boundary we hold throughout, and which is load-bearing for trust: nothing here carries authority because the language model produced it.
It writes and orchestrates software and drafts proofs (§10); it does not compute the invariants, and a proof counts only as text a reader can check. Every invariant is computed by established tools (SageMath, SnapPy, Regina, KnotJob) and independently re-verified; the proofs presented as theorems (§6.1, §6.3) are elementary and fully written out, with their identities checked symbolically; §6.2 is explicitly a sketch with named gaps; and the written proofs C8–C11 have so far been read only by other instances of the language model, not yet by a knot theorist, as each of them states. Nothing is "computed by the AI" in the sense that would matter for trust.
Neither half produces this without the other. "We" throughout means both. Our papers on Zenodo and arXiv name W.S. as sole author and state the collaboration with Claude Code, and who contributed what, in a note on their first page; this report names both as authors. The two forms describe the same division of work and follow the conventions of the two venues. The legal status of an AI coauthor is unsettled and is not claimed here; we name the contribution as authorship-as-fact and leave the formal question to the venue.
We describe a reproducible pipeline that takes the Fourier-series centerline of a physical knot sculpture (a closed space curve, format .fseries) and returns a full invariant fingerprint and a knot-type identification, computed under a strict blind-test discipline (no knot-table lookups in the identification path). We apply the same machinery to a family of constructive generators — an atomic 3D module placed under cyclic (Cₙ) or Schoenflies (Sₙ) symmetry with a closure constraint — which produce parametrized families of knots ("Stanggassinger families"). From this we report:
We also list contributions to open problems posed by others (§4, C1–C12); one of them, a proof of Conjecture R of Di Prisa and Şavk on the symmetry group of Turk's head knots, is our first arXiv paper (arXiv:2610.16600). Throughout we are explicit about evidence status: theorem, measured pattern, or conjecture. Data, curves, and per-knot invariant records are public.
Keywords: experimental knot theory, hyperbolic knots, knot census, amphichirality, finite-type invariants, mutation, fibered knots, mathematical sculpture.
The objects of this paper have a dual origin that is unusual for a knot-theory study.
The first origin is physical sculpture. The first author builds closed-loop stainless-steel knot sculptures. Of the nineteen works discussed here, five exist as hand-welded steel — these span 58 cm and 9 kg to 170 cm and 120 kg —, one (Sundowner) is cut to length but not yet welded, and the remaining thirteen are computed designs that have not been built, with computed extents running up to 334 cm and 450 kg. (Figures give the overall envelope with base included; quoted here is the largest of the three edges, which need not be the standing height. Per-work dimensions are listed on each sculpture page.) Each sculpture is, mathematically, a smooth closed space curve — its centerline. We record that centerline as a truncated Fourier series (the .fseries format, §2.1). A sculpture is a knot; the question "which knot?" is a priori open, because the curve is designed by eye and hand, not from a diagram.
The second origin is constructive. The same aesthetic — an atomic module repeated under a rotational or mirror-rotational symmetry and closed up on itself — can be driven by a generator instead of by hand. Turning the symmetry order, the shear, and the twist produces families of closed curves. These constructive families are the larger part of our corpus and the source of most of the new knots reported here.
The bridge between the two is the identification pipeline (§2): a single computational path that maps either kind of centerline to a knot-type fingerprint, blind to any table.
Three goals:
This began as an experimental-mathematics report, in the tradition of computational discovery followed by honest labeling of evidence status. The pipeline behind it is now a proof tool: the report carries a proved theorem (§6.3), a known construction with its proof written out (§6.1; prior art: Conant–Manathunga 2016), open cases settled with proofs (C4), and proven crossing-number bounds for every novelty candidate (§3.3). The report as a whole is not a proof-paper, because much of what we measure is not yet proved: many structural findings remain measured patterns — verified by computation across a family, but not proven — and a pattern that holds in every case we computed is still not a theorem. We mark every claim as one of:
We are explicit because the alternative — dressing a measured pattern as a theorem — is precisely the failure mode this project guards against. We also document the corrections we had to make to our own corpus (§9): several apparent results were over-claims that measurement later falsified, and we treat that history as part of the method's rigor, not as something to hide.
The mathematics we use is standard (Alexander, Jones, HOMFLY, Khovanov, hyperbolic volume; Stallings fiberedness; Bennequin and Rudolph genus bounds; Goussarov–Habiro/Stanford finite-type theory; Hartley–Kawauchi amphichirality constraints; the Bar-Natan/van der Veen θ invariant; Morton's Hecke-trace HOMFLY). Our contributions are (a) the geometry-to-identity pipeline as a reproducible artifact, (b) the characterization of the constructive families, (c) the census of candidates beyond table range, and (d) scalable computation that reaches where standard tools time out. Several of the open questions below are the subject of correspondence with knot theorists; where a question is genuinely theirs to settle, we say so rather than guess.
A note on the acoustic rendering. Each hyperbolic knot in the corpus is also given an intrinsic tone: the lengths of the short closed geodesics of its complement — a Mostow-rigid invariant — are mapped to audio frequencies on an absolute scale, the systole as the fundamental and longer geodesics as overtones. Rendering a Laplace or length spectrum as sound is itself old, a lineage from Weyl's law through Kac's Can one hear the shape of a drum? (1966) and the Gordon–Webb–Wolpert counterexample (1992); isolated sonifications of knots also exist (parametric torus-knot synthesis [Kasich, NIME 2024]; knots as musical gesture [Mannone, Knots, Music and DNA]).
We are, however, not aware of prior work that renders the hyperbolic length spectrum of a knot complement as an absolute-pitch acoustic fingerprint and uses it to compare knots across a corpus; we present it as a feature, not a theorem. It carries a precise, known limitation: the length spectrum does not determine a hyperbolic knot complement up to isometry — Millichap [arXiv:1406.6033] and Futer–Millichap [arXiv:1609.00748] construct non-isometric knot complements, by mutation, sharing volume and the initial complex length spectrum — so the tone is mutant-blind on its short end, exactly as our corpus shows for the pair 13a_1231 / 13a_1237, while remaining faithful enough that specific complements are spectrally determined (the figure-eight complement; [Garoufalidis–Reid, arXiv:1509.05309]). A knot and its mirror are isospectral and so sound identical; the θ hash distinguishes them.
The coauthorship line names one language-model coauthor, "Claude Code (Anthropic)". That is the convention of this venue and of our deposits, and it is a simplification: the work behind that name is distributed over several model instances of one vendor and, since October 2026, over systems of other vendors used as readers and, in one case, as a contributor. This section records that structure as far as it is documented, so that the standard the report applies to its results (§1.3, §9) also applies to the collaboration itself. The table is deliberately incomplete: it lists what has a written receipt, not what we remember. Responsibility for every sentence of this report remains with W.S.
Instances and roles (state of 5 November 2026).
| Instance | Role | Receipt |
|---|---|---|
| Claude Code (Anthropic), the working sessions; different Anthropic models have driven them over the project's life, the session thread records which | the coauthor named above: generators, pipeline, corpus, proofs as text, this report (§10) | coauthorship statement; §8 (the working protocol); §10 |
| fresh Claude Code instances without shared context | adversarial cold reads of every deposit and of this report before upload; the nine trial runs of the workshop kit (the folder we give away for that reading discipline: workshop-kit.zip with its sha256 on the project's method page, CC BY 4.0) — all nine in our own environment with one tool, none outside; the kit is archived on Zenodo since 6 November 2026 (concept DOI 10.5281/zenodo.24005142, record 24005143), so the zip has a fixed hash and a date | each deposit's README and tool-use section; the kit's README |
| claude.ai (Anthropic, chat), given the public text by W.S. | the critique of v0.22 (16 October 2026): ten points, nine upheld, worked into v0.23 | version history, v0.23 |
| another AI system, given the public text by W.S. | the reading of 24 October 2026: three sentences corrected in v0.25; the system was not recorded | version history, v0.25 |
| DeepSeek (web application, deep-think mode) | in the note on the leading coefficient (Zenodo concept DOI 10.5281/zenodo.22769534): the bijection of version 7 (dated 4 November, published 5 November 2026, record 23972566), found in answer to our question and verified by hand and by machine; parts of the proof of its Proposition 21 in version 8 (published 6 November 2026, record 24004633); a referee pass on that text; and the proposal that led to this section (5 November 2026) | the note's tool-use section and README, versions 7 and 8 |
| ChatGPT (OpenAI) and claude.ai | outside reads of version 8 of the same note before upload (5 November 2026): two findings worked in, one extraction artefact | the note's tool-use section and README, version 8 (names both) |
Not documented, and not reconstructed: which session wrote which sentence of this report, and reads that left no written report. The reason is a property of the instances, not a flaw: no language-model instance can testify to its own history beyond what stands in its context or in an archived object. That is why the record, not the recollection, carries the claim — the same rule the report applies to its mathematics.
External anchors, by type. A reaction counts as an anchor in the sense of priority only when it answers an archived object (a deposit with a DOI, an arXiv posting); a reaction to a mail is correspondence. Since 31 July 2026 every external contact about a deposited result follows its deposit and DOI, so that the object's timestamp is independent of the answer. What is not a deposit — a missing pair in someone's table, a correction to someone's data — goes by mail and is marked as correspondence; the Stoimenow row is such a case. One contact in the table predates the rule (Jabłonowski) and is marked; so does the first mail wave of June 2026, named after the table.
| Date | Counterpart | What happened | Type | Object |
|---|---|---|---|---|
| 2026-07-09 → 07-11 | M. Jabłonowski | our check of his inequality at 14 crossings sent 9 July; reply 11 July; the deposit (DOI 10.5281/zenodo.21389681) followed on 16 July | reaction to a mail — before the DOI-first rule | C1 (withdrawn as a verification in v0.23) |
| 2026-09-22 → 09-23 | A. Zupan | r_s(12n706) = 4 deposited 22 September (concept DOI 10.5281/zenodo.22893039), mail the same day citing it; independent confirmation 23 September | reaction to an archived object | C4 |
| 2026-10-03 | T. Kálmán (10:57) · J. Boninger (12:43) · F. Ben Aribi, A. Guilloux, K. H. Wong (13:45) | results sent, each after its deposit and DOI. Kálmán and Ben Aribi et al.: no reply as of 5 November 2026. Boninger: reply on 26 October, with his own program, to our mail of that day announcing version 2 of the deposit; code comparison exchanged 27 October | correspondence | C8 · C7 · C5 |
| 2026-10-05 → 10-09 | arXiv math.GT endorsement | C9 deposited 5 October (DOI 10.5281/zenodo.23154788) and sent to Di Prisa and Şavk; endorsement granted 9 October 07:37; arXiv's notice names no endorser, O. Şavk wrote the same day that he had provided it, our thanks went to him on 12 October | reaction to archived objects and mails | C9; C12 (arXiv:2610.16600, 12 October) |
| 2026-10-26 | A. Stoimenow | a pair missing from his 15-crossing mutation table (15_13571 / 15_13629) sent by mail; the correction, with our name and a link to our site, on his table page the same day | reaction to a mail — the pair is in no deposit | §2.5, §7 |
| 2026-10-26 → 10-29 | OEIS A163566 | a(9), a(10) deposited 26 October (DOI 10.5281/zenodo.23660288), proposed 27 October, approved 29 October | reaction to an archived object | — |
| 2026-10-30 | J. Brod, arXiv:2610.41996v1 | cites our proof of Takemura's conjecture, arXiv:2610.32269 (found by us on 4 November) | citation of an archived object | C11 |
| 2026-11-01 → 11-03 | OEIS A343358 | a(14), a(15) deposited (concept DOI 10.5281/zenodo.23855791, version 2 on 3 November); not yet submitted to the OEIS as of 5 November 2026 | deposit only | — |
The first mail wave of 15 June 2026 (Bar-Natan and van der Veen, Becerra, Stoimenow) predates the DOI-first rule; it carried findings as text, not deposits. Where a knot theorist answers one of the written proofs C8–C11, agreeing or disagreeing, the evidence tag of that item changes, and the change will appear in the version history, not as a silent edit.
A note on disclosure boundary: this section describes what is computed — inputs, stages, outputs, invariants — and deliberately does not describe the proprietary kinematic mechanism by which crossings are resolved over the camera sequence. That boundary is intentional and is held consistently throughout.
.fseries)The input is a truncated Fourier series of a sculpture's centerline — the space curve threading the steel tube. The file is plaintext: a header followed by blocks of coefficient rows encoding the three coordinate functions of the closed curve. (A symmetry tag in the header, e.g. Z6, records the construction symmetry where known.)
Provenance. A physical square tube → the centerline modeled as a Bézier curve in Blender → exported as a Fourier-series centerline. The .fseries is the canonical record object: identity is computed from the served curve, not from a hand-drawn diagram. A folder without an .fseries is a study and does not count as a work.
Resolution caveat (load-bearing). Truncation order matters. One sculpture (double-6) was first fit with 20 Fourier terms against a dense 411-point blend curve, a ~28% RMS error that smoothed a genuinely complex knot into a spurious simpler one. The correct identity emerged only at higher resolution. The resulting doctrine: Nyquist-respecting sampling and at least two independent resolutions before any knot claim; densify the polyline before fitting. This episode is one reason we trust the corpus only after re-resolution (§9).
A single command runs nine stages on one .fseries, typically a few minutes per knot:
| # | Stage | Output (what) |
|---|---|---|
| 1 | Scanner | signed crossing events with depth ordering, per viewing frame |
| 2 | Tracker | crossings tracked across the frame sequence on camera-independent arc-parameter axes |
| 3 | Resolve | a signed Gauss code + a classicality check |
| 4 | Simplify | crossing-count reduction sweep across frames |
| 5 | Mandatory invariants | the full mandatory invariant list (multiple independent engines), with an internal consistency check |
| 6 | Mandatory-extra | Khovanov, length spectrum, s-invariant, unknotting, Kauffman, bridge number |
| 7 | Enrichments | repair/extension of optional fields in waves |
| 8 | θ + crosscheck | the canonical θ cluster hash (§2.5); then, outside the sealed path, a table-lookup crosscheck |
| 9 | Sort + render | the per-knot record, a diagram (planar-diagram SVG or braid diagram), and a rotatable 3D viewer |
The mandatory invariants are computed by several independent tools and cross-checked against one another (the consistency quartet, §2.4). The final per-knot record carries on the order of 190 invariant fields (more output slots, including bounds and status flags).
No knot-table lookups in the identification path. Rolfsen, KnotInfo, and the snappy census are sealed off from stages 1–7 by an automatic call at the start of every pipeline script: latent table accesses inside the underlying computer-algebra tools are disabled, and for crossing number above 16 the census-based identify() is deactivated entirely (only census-free quantities such as volume and the triangulation isosig remain available). Table lookup is permitted only in the separate stage-8 crosscheck, which reads the finished record and writes a separate file without altering it.
The point is integrity: the knot's identity and its novelty status are derived independently, and only then compared against tables. This forecloses circular "look it up, then confirm it" reasoning, and it is what lets us claim a genuine blind test.
The mandatory list spans twelve sections: (1) diagram/notation; (2) classification and symmetry — including amphichirality, invertibility, reversibility, symmetry type, fiberedness, mutation; (3) algebra — Alexander, Conway, Jones, HOMFLY, Kauffman, determinant, signature, Arf, Seifert matrix, colored Jones, A-polynomial; (4) categorification — Khovanov homology and knot Floer homology with their derived quantities, s-invariant, υ; (5) genus/slice/concordance; (6) Vassiliev/finite-type (v₂, v₃); (7) hyperbolic geometry — volume, Chern–Simons, cusp data, symmetry group, canonical retriangulation isosig, length spectrum; (8) Legendrian/contact; (9) distances; (10) knot group/Wirtinger; (11) link-specific; (12) meta/identification.
Doctrine: every field is attempted on every run. Unreachable fields receive an explicit not_computable status with a note; structurally-empty fields receive not_applicable. Fields are never silently blank. An auto-checked consistency quartet ties four independent computations together: Jones evaluated at i, the Sage determinant, the total rank of knot Floer homology, and |det(V + Vᵀ)| from the Seifert matrix must agree.
We use the Bar-Natan/van der Veen θ invariant as the mandatory canonical hash for clustering and distinctness. It is computed as a value at a rational point (see §2.6), not as a closed symbolic form; the symbolic θ = 0 set (the amphichiral knots) is a true subset of the point-evaluated θ = 0 set, and point evaluation is both consistent and far cheaper.
Known limitation, stated plainly: θ is blind on some mutant pairs. On the Conway/Kinoshita–Terasaka pair and on 13a₁₂₃₁/13a₁₂₃₇, θ is bit-identical, and that blindness is what motivates the separate mutant-separation computation of §7. θ is not mutation-invariant in general: on thirty constructed Conway-mutant pairs — pretzel knots P(2, b, c, d) against P(2, c, b, d), each pair two genuinely different knots (different isometry signatures) with equal Alexander, Jones, HOMFLY and determinant — θ separates all thirty (deposit DOI 10.5281/zenodo.21946255, with 13a₁₂₃₁/13a₁₂₃₇ included there as the counterexample). Which mutants θ sees and which it does not is an open question for us, flagged for outreach rather than claimed as a result. (Correction v0.23: earlier versions called θ "mutation-invariant" and this "a known property of θ" — contradicted by our own deposit of 2026-08-15 and never sourced.)
The methodological core is that the knot is computed over the camera/time sequence — temporal resolution — rather than read off one static diagram. The first author's framing: the temporal machine is "not nearly as elegant as the [symbolic] machine, but with a certain robustness — a duck, but it overtakes any Bugatti." It reaches where elegant diagram invariants hit the exponential wall, and it works directly from the 3D curve rather than from a given diagram. The latter is the genuine asymmetry between this work and symbolic approaches: we start from geometry.
Demonstration (double-6). A genuinely complex sculpture knot — crossing number at most 78, determinant 483 428 169 = 21 987², amphichiral, non-invertible, Z/6 symmetry, hyperbolic volume ≈ 130.15, Alexander degree 36, genus at least 18 — was identified three independent ways: temporal scans at two sampling resolutions plus a direct 3D-projection path, all returning the bit-identical determinant 483 428 169. The same method validates on the trefoil. At 78 crossings the state-sum invariants (HOMFLY, Jones, Kauffman; cost ∼2ᶜ) are not computable; the determinant, the point-evaluated θ, and the temporal resolution carry the identification.
Point-evaluation lever. The θ "cliff" (around 31 crossings for the full symbolic form over a two-variable function field) affects only the closed symbolic form. Point-evaluation over the rationals gets every case in seconds. Measured: the entire machine corpus — including two knots of ∼144 and ∼149 crossings — point-evaluates θ in about 13 seconds total in a single session, versus a 3600-second symbolic timeout on a single 32-crossing knot. The lever is not "work modulo a prime" (that was a net loss at our sizes) but "evaluate at a point at all, instead of building the symbol." Doctrine: for clustering and distinctness, always point-evaluate; reserve the closed symbolic form for small showpiece values. This is also a lesson against a reflex toward elegant closed forms when only a value is needed.
All counts below are quoted from the live corpus database (labeled_at 2026-09-20), which is the single source of truth; any number hardcoded elsewhere in the project is treated as stale.
xyz. The 334 records without a page: the 222 realizations identified as table knots and the 9 unknot degenerations; 88 singleton records whose knot turned out to be a table knot or non-hyperbolic (the Burton retractions of §3.3 among them); 13 further singleton realizations and 2 cluster members of knots that already have a page. 509 + 334 = 843. Records, distinct knots and served centerlines are three counts of one corpus; multiplicity stays in the corpus as evidence (§3.2) and the site shows each knot once.su_* knots reclassified out of the novel count — a connected sum is never hyperbolic, snappy had reported a JSJ-summed volume; 2026-06-24: +33 braid-word knots had been integrated, then 2026-06-27 removed in full — they were braids taken from an external website, not our own constructions, so they were never ours to count; this reverted the then-headline 317→284; 2026-09-04: 17 candidates retracted against Burton's c ≤ 19 census and 82 moved to the unresolved class, headline 279→197; 2026-09-05: 13 of those promoted back via the genus bound c ≥ 2g+1 with HFK-exact genus, headline 197→210; 2026-09-06: two more promoted the same way after the census closure completed the missing knot-Floer data, headline 210→212; 2026-09-09: kf320, a pipeline-failure record completed by hand — determinant 112903 confirmed by two independent computations, hyperbolicity proven by interval arithmetic, determinant proven absent from the complete c ≤ 19 census — was salvaged into the corpus as novel-hyperbolic with span-certified c ≥ 32, headline 212→213; 2026-09-09: seven unresolved knots promoted when the genus route was extended to c ≥ 2g + b − 1 (§3.3), headline 213→220; 2026-09-19: two records that had been counted as two distinct knots proved to be one — identical Jones and HOMFLY, θ ≡ 0 on both sides, isometric complements; they were split only by the ± sign convention of the Alexander polynomial in the distinctness key, which now compares a normal form, headline 220→219; 2026-09-20: 34 knots moved to the new unresolved-alternating class — the alternating prime knots are enumerated through c = 23 (Rankin–Flint–Schermann 2004) and c = 24 (Fontaine 2007), so a proven c ≥ 21 no longer suffices for an alternating knot, headline 219→185).S_24_d81_D6_amphi (26 members, the ping-pong anchor, A5) and S_18_d243_D3_chiral (24 members).singletons_novelty (records outside any S-cluster — a realization count, not a count of novel knots: they are 363 distinct knots, and by class 234 records are novel-hyperbolic, 37 known, 51 known-non-hyperbolic and 66 unresolved; the folder name predates the novelty criteria of §3.3); 9 unknot degenerations (a separate bucket: sweep degenerations, kept as basin-boundary evidence, not knot classes).Each S-cluster is one hyperbolic knot; its members are machine reparametrizations (different shear/twist/helix settings) of the same knot. The parameter→knot map is many-to-one. We therefore key distinctness on a manifold-level tuple — hyperbolic volume (9+ digits), cusp shape, exact algebra (Alexander/Jones/HOMFLY), point-evaluated θ, determinant, signature — and explicitly not on the triangulation isosig, which is a triangulation invariant: different diagrams of the same knot give different isosigs and would over-count.
Multiplicity is kept, never deduplicated: the many realizations of one knot are family and anchor evidence. (A floating-point caveat: within the largest cluster, 26 members, the volume values spread over ∼2.8 × 10⁻¹⁴, i.e. noise in the 14th decimal, not topological separation; agreement is asserted only at the matched precision.)
A family needs only one discovered member. Each family has at least one discovered realization (member); its further members — additional parametrizations that yield the same knot — exist by construction but are, as yet, undiscovered. A one-member family is therefore not "thin": it is a family with a single known realization, its further members reachable by the generator on demand. Reported member counts are discovered so far, not totals.
"Novelty" here has two stages, and since 2026-09 we separate them explicitly. Stage one — lookup. The tables we can actually search cover crossing number ≤ 13 (KnotInfo) and ≤ 14 (the Hoste–Thistlethwaite table shipped with SnapPy 3.3.2: 59 937 knots, none with 15 or 16 crossings, measured 2026-10-17 — earlier versions said ≤ 16); beyond that we match against Maguire's hyperbolic-Jones census (c ≤ 17) and Burton's complete prime-knot census (c ≤ 19, held locally). The Burton passes retracted 30 former candidates in 2026-09 (17 against c ≤ 18 on 2026-09-04; thirteen more on 2026-09-06, when the c = 19 stream, a retriangulation pass and a rigorous closure of the c ≤ 16 layer completed) — among them the sculpture knots trainstation/sundown (= census knot 18ah_5090151) and Spiral (= 18nh_22527179), and three knots at 15-16 crossings that the earlier lookup layer had missed. Stage two — enumeration.
There are two enumeration boundaries, not one, and we got this wrong until 2026-09-20. All prime knots are enumerated through c = 20 (Thistlethwaite, Algebr. Geom. Topol. 25 (2025) 329–344: 1 847 319 428 knots, independently confirmed by Burton). The alternating prime knots, however, are enumerated much further: through c = 23 in the published literature (Rankin, Flint & Schermann, J. Knot Theory Ramifications 13 (2004) 57–150; 25 182 878 921 alternating knots at c = 23, run completed 2004-03-14) and through c = 24 by a later run (Fontaine, Rankin & Flint 2007; OEIS A002864, a(24) = 128 564 665 125).
Thistlethwaite states it in the same paper we cite for the c = 20 count: "This has indeed been accomplished quite dramatically for n ≤ 23 [Rankin et al. 2004]." Neither the c = 20 data nor the Rankin–Flint–Schermann tables are public (Hoste, Handbook of Knot Theory: "not yet publicly available"), so below those lines a knot may be unmatched only because we cannot look it up. A record is therefore a novelty candidate only when (i) it is hyperbolic (volume > 0, confirmed; primality follows from hyperbolicity, since composite knots are never hyperbolic), (ii) it is unmatched against every table above, the Fremlin catalogue, and our θ database, and (iii) it is proven to lie beyond both boundaries — that is, c ≥ 25, or c ≥ 21 together with a proof that the knot is not alternating.
The crossing-number bound comes from one of two independent routes: the Kauffman–Murasugi–Thistlethwaite span bound span V(K) ≤ c(K) — calibrated on all alternating records in the corpus, where span = c must hold exactly and does, with zero violations corpus-wide — or the genus–braid-index bound c ≥ 2g + b − 1 — Seifert's algorithm on a minimal diagram gives a Seifert surface of genus g_D ≥ g with c = 2g_D + s − 1, and the number s of Seifert circles is at least the braid index b (Yamada). The bound is monotone in both g and b, so it stays a lower bound when each enters as a proven lower bound: g is the HFK-exact Seifert genus where knot Floer homology has been computed (it detects genus), otherwise g ≥ ½·span Δ (the Seifert inequality); b is the Morton–Franks–Williams lower bound from the HOMFLY polynomial where available, otherwise b ≥ 2, which holds for every knot proven non-trivial.
(Up to v0.7 only the special case b = 2 with HFK-exact genus, c ≥ 2g + 1, was used.) Whether a knot is alternating is decided from the same theorem, without any external data: span V(K) = c(K) holds exactly for alternating prime knots and drops by at least one for every other prime diagram. So span V(K) equal to the size of a realized diagram proves the knot alternating (and fixes c exactly), while span V(K) strictly below an independently proven lower bound on c proves it not alternating. Three classical sign/size theorems give further exclusions, all applied here: the Alexander polynomial of an alternating link is alternating, i.e. its coefficients satisfy (−1)^(i+j) c_i c_j ≥ 0 (Crowell 1959, Murasugi 1958); the Jones polynomial of a non-split alternating link is alternating in the same sense (Thistlethwaite 1987); and det(L) ≥ c(L) for a non-split alternating link (Bankwitz 1930).
All three are calibrated on every record in the corpus that carries an alternating diagram — 122 of them, zero violations — and together they decided three of the five knots that the span test had left open. In the positive direction we construct an alternating diagram with exactly span V(K) crossings and prove it is the same knot by an isometry of the complements (Gordon–Luecke); this settled the remaining two, leaving one knot genuinely open. 252 distinct knots pass both boundaries: 235 with c ≥ 25, and 17 with c ≥ 21 and proven non-alternating. Two explicit classes hold the rest: 47 unmatched hyperbolic knots without a c ≥ 21 certificate (unresolved, inside the fully enumerated c ≤ 20 range), and 34 with c = 21…24 that are alternating (33) or undecided (1) — unresolved_alternating, inside the enumerated alternating range. For both classes the unmatched status is a statement about our tables, not about the knots. Even for the 252, "novelty" means beyond every existing enumeration, not a theorem of mathematical newness.
Knots that are table-misses but mathematically known are kept out of the novelty count and labeled honestly: T(3,7) (a torus knot, crossing number 14, beyond KnotInfo but classical), and the connected sums. This is why the 43 known-non-hyperbolic distinct knots sit in their own bucket.
These are results a reader can recompute from public data; they anchor the rest. All values are from the live corpus database.
A1 — knots-of-love = 3₁ # 3₁ # 3₁ # m3₁ # m3₁ # m3₁. A six-fold connected sum of trefoils — three right-handed and three left-handed, that is, the square knot three times over. Crossing number 18, determinant 729 = 3⁶, signature 0, volume ≈ 0 (non-hyperbolic), Alexander polynomial (t² − t + 1)⁶ bit-exact, genus 6, fibered, amphichiral, θ ≡ 0. Labeled honestly as a known composite, not a novelty.
(Correction, retained for the record. This anchor previously read as six same-handed trefoils. Determinant and Alexander polynomial are blind to chirality and agree with either reading — which is exactly why the error survived being called "bit-exact". The signature is not blind: it is additive under connected sum, so a same-handed six-fold sum has σ = ∓12. The connected-sum machine of §5 produces precisely that family, with det = 3ⁿ and σ = ∓2n reproduced at the diagram bit level; knots-of-love, with σ = 0, is therefore a different knot from the machine's six-fold product, sharing its determinant and its Alexander polynomial but not its signature. The contradiction stood inside this anchor's own sentence — σ = 0 next to σ = −2n — and went unnoticed until 2026-07-26. An anchor needs at least one chirality-sensitive quantity; det and Δ are not enough.)
A2 — celtic-for-playing ↔ 12n_706. A sculpture and a six-member machine cluster that are one mathematical object. The sculpture record carries distinct_knot_id = 12n_706: crossing number 12, determinant 49, signature 0, volume 13.417 374 391 875 674, D₆, amphichiral; identified as KnotInfo 12n_706. The shared identity has seven realizations (the sculpture plus six machine records). This knot is a ribbon knot, hence smoothly slice, hence g₄ = 0 — see A7. (Correction, retained for the record: double-6 was previously mis-attached to this anchor; it is a different knot — see §9.)
A3 — irrational = T(3,7). A torus knot: crossing number 14, determinant 1, signature −8, volume 0, symmetry "D₂₁", non-amphichiral; Alexander (t²¹ − 1)(t − 1)/((t³ − 1)(t⁷ − 1)) bit-exact, genus 6 = (p−1)(q−1)/2, fibered. Beyond KnotInfo (c = 14) but classically well-known; not a novelty. (The signature is reported as −8 under the project's truth-faithful global mirror convention; an older file shows +8 as "mirror T(3,7)" — the live value governs.)
A4 — nitro ↔ machine → S_24_d961_D6_amphi (the four-modality anchor). Crossing number 24, determinant 961 = 31², signature 0, volume 32.262 943 938 951 26, D₆, amphichiral, θ-trivial; full agreement of Alexander, Jones, HOMFLY, Δ(22/7), and volume (14 digits) between the sculpture and three machine realizations (four realizations total). Outside KnotInfo, absent from Burton's complete census through c = 19, no Fremlin match. It is not a novelty candidate: its proven lower bound is c ≥ 20 (absence from the census), so it lies in the unresolved class inside the enumerated range c ≤ 20 (§3.3); 24 is the crossing number of our diagram, an upper bound. Unlike A2, which points into a public table, this identity is between two of our own computations. (Correction v0.23: earlier versions called this "the first sculpture↔machine match inside genuine novelty" — written before the c ≥ 21 criterion of v0.5 and never updated.) Alexander x⁸ − 16x⁷ + 90x⁶ − 224x⁵ + 299x⁴ − 224x³ + 90x² − 16x + 1; HFK genus 4.
A5 — ping-pong ↔ S_24_d81_D6_amphi. The largest S-cluster (26 realizations including the sculpture). Crossing number 24, determinant 81 = 3⁴, signature 0, volume 24.351 352 283 060 53, D₆, amphichiral, θ-trivial. Alexander/Jones/HOMFLY/Conway each a single value across the cluster and bit-matching the sculpture; Δ(22/7) bit-matched.
A6 — sculpture ≡ sculpture: trainstation ≡ sundown. Two independently designed works that are the same knot — trainstation exists as hand-welded stainless steel, sundown is a rendered design that has not been built: crossing number 18 (proved minimal, reduced alternating diagram, Tait/KMT), determinant 3969 = 63², signature 0, volume 33.600 846 951 271 4(5), D₆, amphichiral, θ = 0; Alexander and Jones bit-equal, volume to 14 digits. The first sculpture≡sculpture identity in the corpus. (This corrected an earlier Frankenstein reading of trainstation as a different knot; see §9.)
A7 — four sculpture records are ribbon knots; their smooth 4-genus is 0. Two distinct knot types in the corpus are ribbon knots: 12a_1019 (the sculptures double-3, always-a-pleasure, proxima-b) and 12n_706 (celtic-for-playing). Both appear in C. Lamm, The search for non-symmetric ribbon knots (arXiv:1710.06909v2, 2019), Table 1 — "a list of all 137 prime ribbon knots with crossing numbers 11 or 12" — and both are given there as symmetric unions: 12a1019 with partial knot 7₆ and axis parameters (1, −1) (appendix p. 20), 12n706 with partial knot 7₁ and axis parameters (1, −1) (appendix p. 18). A symmetric union bounds a ribbon disk by construction (Kinoshita–Terasaka 1957; the fold along the symmetry axis, Lamm Def. 6.1, yields a band composed only of strips, ends, twists and ribbon singularities — Lamm Prop. 5.1, which credits Aceto; we hold Lamm's paper, not Aceto's, and cite only what we have read). Ribbon ⇒ smoothly slice ⇒ g₄ = 0, smooth and topological.
Cross-check, computed and not cited. For a symmetric union, det = det(K±)². We computed det(7₆) = 19 and det(7₁) = 7 as the torsion of H₁ of the 2-fold cyclic cover (SnapPy; the method calibrated on the counter-case 4₁ → 5). Then 19² = 361 and 7² = 49 — exactly the determinants our pipeline had computed blind from the .fseries, before any of this was looked up. What this test does and does not do, stated exactly: our determinant was one of the keys the table match itself used (queried_c_det_sig = [12, 361, 0]), so the agreement is not a second, independent identification of the sculpture. What it does test is the cited decomposition: given that the knot is 12a1019, Lamm's appendix line predicts a partial knot of determinant 19, and a wrong line there would have produced a mismatch. It is a check on the citation chain, and it would have caught an error in it.
What this changes in our records. Our slice verdict is an obstruction search: it can refute sliceness, never establish it. For these knots it had found every necessary condition satisfied (τ = s = σ = Arf = 0, determinant a perfect square) and correctly returned "undetermined", with the smooth 4-genus bracketed as 0 ≤ g₄ ≤ g₃. The literature collapses that interval to a point: for double-3, g₃ = 4 but g₄ = 0 — a surface of genus 4 is needed in ordinary space, none at all in the 4-ball. double-3 stands as welded stainless steel. Propagated across the corpus by knot identity rather than by record name, this moves 39 records (32 of 12a_1019, 7 of 12n_706) from "undetermined" to slice with a receipt; with knots-of-love (A1) and mirror — both our own K # m(K) arguments — the corpus now holds 41 records with a proven slice verdict. For mirror = 5₂ # m5₂ the handedness is fixed by the signature, the one chirality-sensitive quantity here: σ(5₂) = −2 and σ(m5₂) = +2 sum to the recorded 0, whereas two same-handed summands would give −4. 5₂ is invertible (computed), so m(5₂) is its concordance inverse and the fold construction applies.
Scope, stated plainly. This is a literature receipt for a property of knots we identified blind — not a disk we constructed ourselves. The blind-test doctrine (§2.3) is not violated: nothing was identified by lookup. Until we build and check the band presentation ourselves, Lamm is the source and is named as such, in the records, on the website, and here.
A8 — two of seven. Lamm, Question 3.2, lists the known prime strongly-positive-amphichiral knots with at most 12 crossings: 10₉₉, 10₁₂₃, 12a427, 12a1019, 12a1105, 12a1202, 12n706. Two of those seven occur in this corpus as sculptures, arrived at from the sculptor's side without the knot being chosen in advance. Both entered that list only through Seeliger's search and Lamm's later diagrams; before that, three were known. (The one case Question 3.2 left open is settled: Lamm, arXiv:2310.05106v2 (12 Dec 2023), tabulates the prime strongly positive amphicheiral knots up to 16 crossings and shows in §2 that 12a435 is not strongly positive amphicheiral. The list of seven is therefore complete.)
External contributions (first, dated, verifiable). Results contributed to open problems posed by others (C1–C12), each recomputable from public data or written out, and each deposited for a citable, non-editable timestamp on Zenodo; C12 is also on arXiv. C1–C3 share one deposit — DOI 10.5281/zenodo.21389680 (concept DOI, https://doi.org/10.5281/zenodo.21389680; version 1 is 10.5281/zenodo.21389681, published 2026-07-16, CC-BY-4.0); the later ones carry their own DOIs below.
C1 — Jabłonowski's candidate inequality sp_a(P) ≤ sp_a(F) at 14 and 15 crossings — computed, but withdrawn as a verification until the tables are recomputed. The inequality sp_a(P) ≤ sp_a(F) (the a-spread of the HOMFLY-PT polynomial is at most that of the Kauffman polynomial), listed as the open candidate (c7) in M. Jabłonowski, Integer Knot Invariants: Inequalities, Computations, and Open Problems (arXiv:2605.22652v3, May 2026), was verified there only up to 13 crossings. Under the paper's half-a-spread convention (calibrated against 12n296/12n550 in the KnotInfo NewDB) we computed it for all 46,972 knots at c = 14 and all 253,293 at c = 15 (253,281 with knotpy; the 12 on which knotpy fails, on 2026-09-18 with an independent Kauffman implementation), and the stored tables show zero violations (minimum slack 2 and 4).
That is not yet a verification. A validity check on 1,600 random rows (2026-09-18) found 6 in which knotpy had silently returned a Kauffman polynomial with unresolved symbols, so the stored a-spread is wrong there; a violation could hide in such rows. We withdraw the claim of a complete verification until the tables are recomputed with a validity check on every row. (Correction v0.23. The deposit cited above states the earlier claim; it will receive a corrected version. Its c = 15 table has 253,281 rows; the 12 missing knots satisfy the inequality, with slack 6 to 8.) Reported to the author 2026-07-09; positively acknowledged 2026-07-11.
Version 4 of the paper (2026-08-12) records this check in its Remark 3.1 and proves the inequality for every alternating knot (Proposition 3.14). Version 5 (2026-09-15) no longer lists the inequality, and its label (c7) there denotes a different comparison; our check concerns sp_a(P) ≤ sp_a(F) as stated above, not the (c7) of version 5. Columns of the deposited tables: knot, spF_a, spP_a, slack, c7_holds (c7 in the numbering of versions 3 and 4).
C2 — Knot 15331 (D. H. Fremlin's knot problem). Run through the identification pipeline: sharper bounds and a short primality/hyperbolicity argument (hyperbolic ⇒ prime). D. H. Fremlin, a mathematician, publicly credited the first author by name on his Knot-15331 catalogue page (note dated 24.5.26) and adopted the sharpened statement "…confirmed prime, crossing number at most 14, fully amphichiral," tightening his prior "crossing number at most 16" and "may be amphichiral." Identity chain: Knot 15331 = our orbit = 14n_9732 (determinant 25, volume 14.135 376 574 931 405).
C3 — A. Hizume's precessing-ellipse / Farey atlas. Hizume's construction threads the Farey sequence into torus knots. Taken across the whole family — closed at Δ = q/p for 28 ratios (3/2 … 13/6) — each resulting space curve was handed blind to the pipeline and identified, with no table lookup, as exactly T(p,q) (determinant and Alexander degree matching the closed-form torus values), each with its Jones polynomial and its standard closed braid (σ₁…σ_{p−1})^q. The citable contribution here is the blind verification and systematization into an atlas — not the underlying construction, which is Hizume's. (Crossing number exact for 16 of the 28; "≈ formula" for the rest, a caveat carried in the atlas itself.)
C4 — Symmetric ribbon numbers: one open case settled, a sculpture knot, and a second one not yet verified. S. Akash, C. Corrado, J. Placke, S. Sanketh, M. Starns, A. Timothy and A. Zupan, Symmetric ribbon numbers of low-complexity knots (arXiv:2606.02390v1, June 2026), leave four of the prime non-alternating symmetric-ribbon knots with 12 crossings undetermined in their Table 3: 12n23, 12n49, 12n504 and 12n706. We settle one. r_s(12n706) = 4. The labeled knotoid diagram with arc word 12314234 and twist labels {1: −1, 2: 0, 3: −1} has four crossings; under the correspondence of the paper's Lemma 2.3 each crossing becomes one ribbon singularity of a symmetric ribbon disk, so r_s ≤ 4.
The other direction is in the paper: r_s ≥ r (Lemma 2.2), with r(12n706) = 4 in Table 3. 12n706 is the knot of the sculpture celtic-for-playing (anchor A2). r_s(12n504) = 5 — not verified; withdrawn as a result in v0.24. Arc word 1232451543, twist labels {1: −3, 2: 1, 3: 1, 4: −1}: five crossings, and the table's lower bound is 5 (det = 121 > 49 also excludes r_s = 4 by the paper's Corollary 5.1). But an independent check of the deposited diagrams on 2026-10-17, written without our construction code, found that in both 12n504 witnesses one of the five singularities violates the "global consistency" that the paper's Lemma 2.3 requires; with that singularity made consistent the boundary is no longer 12n504.
So r_s ≤ 5 does not follow from Lemma 2.3 as written. It would follow if the two choices of singularity at a crossing give the same piece of a modular ribbon disk — plausible, and supported by 120 test constructions, but not checked against the definition by a knot theorist. Until then r_s(12n504) = 5 is a candidate, not a result. The same check confirmed all four 12n706 witnesses against Lemma 2.3 as written. In both cases the knot we build is identified with the target knot by an isometry of the complements verified in interval arithmetic (SnapPy isometry_signature(verified=True)), not by comparing polynomials.
A. Zupan has confirmed r_s(12n706) = 4 independently, from the DT code we sent him (personal communication, 23 September 2026). The 12n504 witness has since been sent to him as a DT code (25 September 2026); it is not confirmed, and in view of the finding above it must not be relied on. Along the way we computed the set R^s_5, which the paper calls "too large to compute via our methods": 145 Alexander polynomials, from 3 328 constructions over the 26 reduced singular arc diagrams with five crossings that we enumerated (R^s_2, R^s_3 and R^s_4, with 27 polynomials, are reproduced exactly).
Two controls hold: every knot of Table 3 with proven r_s = 5 carries its Alexander polynomial in our set (17 of 17), and the largest determinant in the set is exactly 169, the bound of the paper's Theorem 1.6, which is proven independently of our construction. What this does not show: the constructor that turns a labeled knotoid diagram into a knot is calibrated against the paper's tables, not proven correct, and no disk has yet been drawn and checked by hand; the two values rest on the witness objects and their verified isometry. That our list of 26 five-crossing arc diagrams is complete is not proven either, so R^s_5 is stated as computed, not as complete. Deposited: DOI 10.5281/zenodo.22893039 (https://doi.org/10.5281/zenodo.22893039, published 2026-09-22).
C5 — The Andersen–Kashaev volume conjecture for 220 knots beyond the 14-crossing census of Ben Aribi, Guilloux and Wong. Evidence status: theorem, by a published corollary applied to certified computations; the theorem and the method are not ours. J. E. Andersen and R. Kashaev built a TQFT from quantum Teichmüller theory (arXiv:1109.6295) and conjectured that its partition function on a hyperbolic knot complement decays at a rate given by the hyperbolic volume. F. Ben Aribi and K. H. Wong (arXiv:2410.10776v2, 2026) introduced FAMED ideal triangulations and proved: "Corollary 1.9. Conjecture 1.5 is true for all FAMED geometric triangulations", Conjecture 1.5 being their formulation of the Andersen–Kashaev volume conjecture. Ben Aribi, A. Guilloux and Wong (arXiv:2512.17437, FAMED by computer: proving the Andersen–Kashaev volume conjecture for 42,000 knots) then found such triangulations by computer for all but 6 of the 42 123 hyperbolic knot complements of the Hoste–Thistlethwaite census with at most 14 crossings and at most 23 tetrahedra. Their census stops there, at the threshold of 24 tetrahedra; our literature search to 25 September 2026 (the arXiv full-text search for "FAMED triangulation" returns three papers) found no case beyond it; rechecked on 3 October 2026, their paper is still at its first version and the project page lists nothing beyond it.
We applied their method to the hyperbolic distinct knots of our corpus and establish the conjecture for 220 knots with 216 distinct complements. None of them lies in the census they covered: 160 have a proven crossing number c ≥ 20 (§3.3), so they lie outside the whole Hoste–Thistlethwaite census (up to 16 crossings); the other 60 were checked with SnapPy's census identification on several geometric retriangulations each, finding none — a check first calibrated on 300 hyperbolic census knots, all found again after randomising their triangulations. SnapPy's table holds the knots with at most 14 crossings, so for these 60 only that range is covered; whether some are Hoste–Thistlethwaite knots with 15 or 16 crossings was not checked. (Correction v0.24: earlier versions called all 220 "outside the Hoste–Thistlethwaite census" and stated the bound as c ≥ 15.) The triangulations we use have 8 to 56 tetrahedra; 106 of them have more than 23. The chain, for every knot:
FAMEDexploration) proposes an ordered triangulation, searching over retriangulations of the knot complement. The verdict is not the authors' code: our independent implementation of their Definition 1.1, conditions (1)–(4), decides it in exact rational arithmetic, on an ordered triangulation of the same complement — for the 87 larger cases exactly the one the search produced, for the others one found by enumerating all orderings — with respect to the homological (preferred) longitude. The checker was tested against itself: flipping one tetrahedron's orientation sign, swapping two face matrices, or substituting the meridian for the longitude makes it reject every one of five FAMED triangulations (15 of 15 perturbations rejected).verify_hyperbolicity). The authors' code tests only for a strict angle structure, which is weaker; we do not rely on it.Among the 220 are 67 twisted torus knots T(p, q; 2, s) (proven lower bounds on the crossing number from 26 to 150) and ten odd pretzel knots P(p, q, r) with 3 ≤ p < q < r ≤ 13. Six sculpture records are among them as well: ping-pong (A5), nitro (A4), trainstation and sundown (A6, one knot), cosmic-love and spiral. Of these, only trainstation exists in stainless steel; the others are rendered designs.
What is ours, and what is not. The conjecture's proof for FAMED geometric triangulations is Ben Aribi–Wong's, and the idea of proving it census-wide by computer is Ben Aribi–Guilloux–Wong's. Ours are the knots — all outside the census they covered — the independent FAMED check, the interval certificates on the same triangulation, and the meridian computation. The objects and certificates are deposited with a citable, non-editable timestamp on Zenodo — DOI 10.5281/zenodo.22975395 (https://doi.org/10.5281/zenodo.22975395, first version with 213 knots published 2026-09-26, version 2 with 220 knots published 2026-10-03 as DOI 10.5281/zenodo.23120187, CC-BY-4.0): for every knot the ordered triangulation, the FAMED verdict, the interval certificate, the meridian data and the diagram, with a script that recomputes all of it (220 of 220 recomputed from the deposited data alone). The result has been sent to the authors (3 October 2026). A caveat on its lifetime: T. Liu, S. Ming, X. Sun, B. Wu and T. Yang (arXiv:2608.16560, August 2026) prove a volume conjecture for a related Turaev–Viro-type invariant for every geometric ideal triangulation, and describe their functions as essentially those predicted to appear in the Andersen–Kashaev partition function. If that bridge is built, lists of individual cases such as ours may be superseded.
C6 — The Vol-Det conjecture for every alternating hyperbolic knot with at most 19 crossings. Evidence status: theorem for each of the 51 280 967 knots, by a published volume bound and exact or certified computation; the conjecture and the bound are not ours. A. Champanerkar, I. Kofman and J. Purcell conjectured that every alternating hyperbolic knot K satisfies vol(K) < 2π log det(K) (arXiv:1411.7915); the K3 problem list (Baykur, Kirby, Ruberman, 2026) carries it as Problem 1.29(a). It is proven for several infinite families — among them 2-bridge knots and alternating 3-braids (S. D. Burton, arXiv:1704.02344), further families by number-theoretic methods (Champanerkar–Kofman–Lalín, arXiv:1805.05490), and highly twisted alternating links (Egorov–Vesnin, arXiv:2411.11711) — and it was verified by computer for all alternating knots with at most 16 crossings. Egorov and Vesnin cite B. A. Burton's census of prime knots through 19 crossings, in which 51 280 967 knots are alternating and hyperbolic, and still give 16 crossings as the verified range; our literature search to 25 September 2026 found no verification beyond it.
We checked all 51 280 967 alternating hyperbolic knots of Burton's census — 50 789 647 of them with 17, 18 or 19 crossings — and the conjecture holds for every one. The chain, for every knot:
verified=True); the decision is that 2π log det − vol, as an interval, lies entirely above 0; its smallest lower end over the 72 is 21.29, and the largest ratio vol / (2π log det) among them is 0.563. What was tight in these cases was the bound, not the knot.Controls. The determinants agree with those of spherogram on a sample of 1 269 census knots (4 to 19 crossings), and a deliberately wrong coloring rule is caught. Our bipyramid volumes reproduce Adams' own table entry for entry. The face structure (c + 2 faces, 4c edge-sides, no monogon) was recomputed for all 51 280 967 knots and matched the first run bit for bit. The range c ≤ 16 reproduces the published verification: 491 320 knots. The determinants of the whole census took under twelve minutes on the card.
What this does not show. Hyperbolicity, primality and the crossing number of each knot are taken from Burton's census, not re-proved here. The general conjecture is untouched; the census ends at 19 crossings.
The deposited note also checks four conditions on the Alexander polynomial for all 51 280 976 prime alternating knots of the census with at most 19 crossings (the 51 280 967 hyperbolic ones and the nine torus knots T(2, c)): Fox's trapezoidal conjecture in its strict form, log-concavity of the coefficients, Hoste's conjecture that every zero has real part > −1 (tested exactly, by an integer Routh–Hurwitz test), and Conjecture 5.11 of Hirasawa and Murasugi on the width of the plateau. All four hold for every knot. The published checks by crossing number that we found for these four go no further than 16 crossings. In particular, given the completeness of the census, a counterexample to Hoste's conjecture among prime alternating knots needs at least 20 crossings. Deposited, as a note with the code, the summaries and logs of all runs and per-knot files for the exceptions: DOI 10.5281/zenodo.23156040 (https://doi.org/10.5281/zenodo.23156040, published 2026-10-05, CC-BY-4.0).
C7 — Boninger's Conjectures 1.1, 1.6 and 1.7 for every alternating hyperbolic knot with at most 19 crossings. Evidence status: a finite computer check with no counterexample; the polynomial and the conjectures are not ours. J. Boninger (arXiv:2608.28484v1, August 2026) associates to every alternating link a four-variable polynomial P_K(x, y, z, w), built from the spanning trees of the Tait digraph, which specializes to the Alexander polynomial. He conjectures that it is symmetric, P(x, y, z, w) = P(y, x, w, z) (Conjecture 1.1), that its support is M-convex (Conjecture 1.6), and that it is log-concave in the sense c_α² ≥ c_{α+e_i−e_j} c_{α−e_i+e_j} (Conjecture 1.7), and he verified all three for the alternating knots with at most eleven crossings. We checked all three on all 51 280 967 alternating hyperbolic knots of Burton's census — 51 280 409 of them with 12 to 19 crossings — and found no counterexample. The number of knots checked equals the number of census entries for every crossing number; it is the same set as in C6.
The chain: P is implemented twice from Boninger's definitions — directly, and through the dual spanning tree — and a C kernel performs the same enumeration for the large runs; the three agree on every knot compared. Boninger's printed P(4₁) is reproduced exactly, and his printed P(8₁₀) under his mirror property (iv) (spherogram's 8₁₀ is the mirror image of his). His Theorem 4.2 (independence of root and basepoint), Proposition 4.4 (specialization to the Alexander polynomial), Proposition 4.3 (homogeneity of degree ±σ) and P(1, 1, 1, 1) = det hold on 40 knots with 10 crossings, and each of the three conjecture checkers fires on a hand-made violation. The census lists each knot once up to mirror image; mirroring and reversal act linearly on the exponents and preserve all three conjectures — measured, not only argued, on 2 092 polynomials, of which 1 429, 644 and 953 violate Conjectures 1.1, 1.6 and 1.7.
One convention. Boninger fixes left and right ("counter-clockwise around the basepoint") only in his figures. Switching it exchanges z and w, which leaves all three conjectures unchanged. In our implementation this switch flips under mirroring: with the same setting on a knot and on its mirror image, property (iv) fails for 38 of 523 census knots with 10 to 13 crossings, always by exactly exchanging z and w. Every P we computed is Boninger's up to z ↔ w; a comparison with his code would settle the convention itself.
What this does not show. Links with two or more components, composite knots and the torus knots T(2, n) were not checked; hyperbolicity and primality are taken from Burton's census. A finite check is not a proof. Deposited, with the code, the per-block results and the log of every check together with its countercase: DOI 10.5281/zenodo.23118923 (https://doi.org/10.5281/zenodo.23118923, published 2026-10-03, CC-BY-4.0). The result has been sent to the author (3 October 2026).
C8 — The Alexander plateau of a two-bridge knot is carried by one coherent twist region. Evidence status: a written proof, not refereed — read adversarially several times by independent instances of Claude Code, not yet by a knot theorist — and tested by machine on finite ranges. Let K be a two-bridge knot whose Alexander polynomial has a central plateau of half-width m ≥ 1, that is, 2m + 1 equal central coefficients in absolute value. Then the reduced alternating (Conway) diagram of K has a coherently oriented twist region in which a single crossing change gives a two-bridge knot K′ with c(K′) = c(K) − 2, plateau half-width m − 1 and |σ(K′)| = |σ(K)| − 2, except for the trefoil, where K′ is the unknot. This is a converse-type statement, for two-bridge knots, to the twist step of S. Azarpendar, A. Juhász and T. Kálmán (arXiv:2406.08662). The proof is an induction over Hartley's moves, using his bottom sequences as presented by W. Chen (arXiv:1712.04993); as a by-product it gives a written-out proof of Chen's theorem that the Hirasawa–Murasugi bound holds for two-bridge knots and links (in the form Z ≤ |σ|), whose induction step for even length is left as "similar" in the arXiv version of Chen's proof.
What is new, as far as we could find, is the descent when the plateau is carried by a deeper chain of Hartley's move T1, which does not reduce to the top chain by Schubert's symmetry (example N(39/14): up to mirror image the only plateau knot with at most nine crossings of which no representative has the plateau on its top chain), the reading of such a chain as a coherently oriented twist region, and, for a deeper chain, c(K′) = c(K) − 2. The walk description of the Alexander coefficients is Hartley's, Minkus' and Hoste's; the bound is Chen's. Beyond two-bridge knots, in Burton's census: all 20 158 hyperbolic alternating prime knots with at most 19 crossings and m = 1 admit such a crossing change, counted up to mirror image. A supplement tests Corollary 3.16 of Azarpendar–Juhász–Kálmán on the 2 120 sharp alternating prime knots with a plateau up to 19 crossings: its conclusions hold for all of them, 124 of them outside its hypothesis.
Deposited, with a pure-Python script that recomputes the machine checks: DOI 10.5281/zenodo.23117903 (https://doi.org/10.5281/zenodo.23117903, published 2026-10-03, CC-BY-4.0). The write-up has been sent to T. Kálmán (3 October 2026); the journal version of Chen's paper has been requested from its author.
C9 — Hoste's condition for every Turk's head knot (Problem H of Di Prisa and Şavk). Evidence status: a written proof, not refereed — read adversarially by independent instances of Claude Code, not yet by a knot theorist — with machine checks of its identities. A. Di Prisa and O. Şavk, Turk's head knots and links: a survey (arXiv:2409.20106v3; Expo. Math. 44 (2026), 125777), ask in their Problem H to verify Hoste's condition for all Turk's head knots; the survey records it for p = 3. We show that every zero z of the Alexander polynomial of every Turk's head knot Th(p, q) satisfies Re z > −1, and that every non-real zero lies on the unit circle.
The proof is an exact factorisation of det(I − x S(σ₁σ₂⁻¹σ₃⋯σ_{p−1}^{±1})) for the reduced Burau representation S into ⌊(p−1)/2⌋ quadratic factors in t, whose middle coefficients involve c_j = 4 cos²(πj/p), and possibly one linear factor, followed by an elementary root estimate. The product-and-recursion structure is already in S. Blackwell, A. Das, S. Mayer, L. Moyar, F. L. Quraishi and R. Stees (arXiv:2506.17889); the explicit factors and the root location are what we add. Checks (the proof does not depend on them): the factorisation exactly for 3 ≤ p ≤ 11 and modulo a prime at random points for every p ≤ 200; the resulting Alexander polynomials against SageMath's Seifert-matrix computation for all 22 Turk's head knots with 3 ≤ p ≤ 8 and 2 ≤ q ≤ 7. Deposited, with the scripts and their outputs: DOI 10.5281/zenodo.23154788 (https://doi.org/10.5281/zenodo.23154788, published 2026-10-05, CC-BY-4.0).
C10 — The degree of the Alexander polynomial of twisted torus knots, for every s. Evidence status: a written proof, not refereed — read adversarially by independent instances of Claude Code, not yet by a knot theorist — with machine checks. For the twisted torus knot K(p, q; r, s), the closure of δ_p^q δ_r^{rs} with 0 < q < p, gcd(p, q) = 1, 2 ≤ r < p and s ≠ 0, we give a closed formula for the degree (highest minus lowest exponent) of the Alexander polynomial, valid for every s: deg Δ = 2 max_m (d_m + m r s) − (q + 1)(p − 1) − (r − 1) r s, where (d_m) is the degree sequence of our earlier note on the leading coefficient (DOI 10.5281/zenodo.22769534).
For s < 0, K. Park and Adnan (J. Knot Theory Ramifications 34 (2025), 2550038; arXiv:2411.13003) determined it in two regimes. The highest exponent comes from the earlier note, whose two theorems used here are re-proved in an appendix; the lowest follows from the symmetry of Δ, whose exact centre is fixed by the exponent sum of the braid. Checks (the proof does not depend on them): against SageMath's Seifert-matrix Alexander polynomial in 640 of 640 cases with p ≤ 8; against the graded Euler characteristic of knot Floer homology (Z. Szabó's program through SnapPy) in all 27 527 evaluated cases with 3 ≤ p ≤ 14 and s < 0; against the two families of Adnan and Park (arXiv:2603.06316; 169 of 169 and 29 of 29). Deposited, with the scripts, data and outputs: DOI 10.5281/zenodo.23155383 (https://doi.org/10.5281/zenodo.23155383, published 2026-10-05, CC-BY-4.0).
C11 — Takemura's conjecture on the Conway polynomials of Th(p, q) and Th(q, p), for all p and q. Evidence status: a written proof, not refereed — read adversarially by independent instances of Claude Code, not yet by a knot theorist — with machine checks of its identities. A. Takemura (J. Knot Theory Ramifications 25 (2016), 1650082) conjectured that the Conway polynomials Σ a_ℓ z^ℓ of the Turk's head link Th(p, q) and Σ b_ℓ z^ℓ of Th(q, p) satisfy a_ℓ = b_ℓ for ℓ ≡ 0, 1 (mod 4) and a_ℓ = −b_ℓ for ℓ ≡ 2, 3 (mod 4), and proved it for q = 2, 3; the survey of Di Prisa and Şavk lists it at Problem E (§6.4).
We prove it for all p, q ≥ 1, knots and links. Equivalently, ∇_Th(q,p)(z) = c · ∇_Th(p,q)(iz) with c = 1 if gcd(p, q) is odd and c = −i if it is even. The proof combines a product formula ∇_Th(p,q)(z)² = ∏_{j=1}^{p−1} ∏_{k=1}^{q−1} (z² + 2 cos(2πk/q) − 2 cos(2πj/p)), which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid. Checks (the proof does not depend on them): the sign rule for all 28 pairs 2 ≤ p ≤ q ≤ 8, knots and links; all 29 Conway polynomials listed by Takemura reproduced; the identity exactly over Q(i) for all 81 pairs 2 ≤ p, q ≤ 10; the product formula exactly for all 64 pairs 2 ≤ p, q ≤ 9. Deposited, with the scripts and their outputs: DOI 10.5281/zenodo.23183920 (https://doi.org/10.5281/zenodo.23183920, published 2026-10-06, CC-BY-4.0).
C12 — The symmetry group of Turk's head knots (Conjecture R of Di Prisa and Şavk). Evidence status: a written proof, posted on arXiv, not refereed — read adversarially by independent instances of Claude Code, not yet by a knot theorist. For p, q ≥ 3 with gcd(p, q) = 1, the symmetry group of the Turk's head knot Th(p, q), the closure of (σ₁σ₂⁻¹σ₃σ₄⁻¹⋯σ_{p−1}^{±1})^q, is the dihedral group of order 4q. This proves Conjecture R of A. Di Prisa and O. Şavk (Turk's head knots and links: a survey, arXiv:2409.20106v3, §6.3) for q ≥ 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram, and the standard diagram of Th(p, q) has no non-trivial flypes and exactly 4q symmetries. Posted on arXiv as arXiv:2610.16600 (math.GT, version 1 of 12 October 2026, https://arxiv.org/abs/2610.16600), our first arXiv paper; deposited with the scripts of its computational checks as DOI 10.5281/zenodo.23234787 (concept DOI; first version 2026-10-08).
The constructive generators place an atomic module (a braid block, or a 3D module) under a symmetry and close it up. We characterize what each knob does. Below, every law is tagged. We computed the braid-word families directly from the braid word in a computer-algebra system; the geometric module families run through the full pipeline.
Building K₁ # K₂ as adjacent braid blocks reproduces the classical laws at the diagram bit-level: determinant multiplicative (3₁ⁿ → 3, 9, 27, …, 729 = 3ⁿ), Alexander multiplicative (Δ = Δ(3₁)ⁿ bit-identical), genus additive (1, 2, 3, 6 = n), signature additive (−2n for like chirality). Notably, the granny (3₁ # 3₁, signature −4) and the square (3₁ # m3₁, signature 0) are separated by signature where determinant (9 = 9) and Alexander (identical) fail. (Status: the #-laws are classical theorems, here confirmed by construction; θ-under-# is open and is a question we have put to Bar-Natan/van der Veen.)
The 3-strand family (σ₁σ₂⁻¹)ⁿ, crossing number 2n, 3 ∤ n, is amphichiral for every n: signature 0 and hyperbolic-complement isometry to the mirror (the sufficient condition, not just the necessary signature 0). Determinant ≡ 1 (mod 4) and a sum of two squares (Hartley–Kawauchi/Goeritz) — not, as we first wrote, a perfect square: 4₁ has determinant 5 = 1² + 2². Measured: n = 2 → 4₁ (det 5, vol 2.030); n = 4 → 8_18 (det 45, vol 12.351); n = 5 → 10_123 (det 121, vol 17.086); n = 7, 8, 10, 11 → hyperbolic, det 841, 2205, 15125, 39601, volumes to 43.12. These are the classical Turk's-head knots Th(3, n) and not novelties: with c = 2n = 14, 16, 20, 22 they lie inside the enumerated ranges — n = 7 is 14a_19470 and n = 8 lies in the Hoste–Thistlethwaite census (c ≤ 16), both labeled known in the corpus; n = 10 lies in the all-knot range c ≤ 20, and n = 11 is alternating with c ≤ 24. All seven determinants equal L₂ₙ − 2, with L the Lucas numbers (L₄ = 7, …, L₂₂ = 39 603), as a reader (an AI system) pointed out. (Status: amphichirality proven per instance via complement isometry; the sum-of-two-squares form and det = L₂ₙ − 2 are measured on seven cases. Correction v0.23: earlier versions called n ≥ 7 "novel".)
A closed 3-braid V(k) = Uₖ · Ūₖ, crossing number 40k + 4, with perfect-square determinant that is not an artifact of compositeness: V₁ (k = 1, c = 44) has determinant 103 041 = 321² and volume 46.074; V₂ (c = 84) has determinant 82 817² and volume 93.124; both amphichiral by complement isometry. The pre-registered adversarial counter-hypothesis — that V is a disguised connected sum, making the square trivial — was falsified: 103 041 ≠ 63² (the control summand's determinant squared), and V₁ is hyperbolic, hence prime (connected sums are never hyperbolic). (Status: prime + amphichiral + square determinant proven per instance; series behavior measured at k = 1, 2.) This is a graduated sibling of the nitro anchor.
A family tw_k whose companion is built from the lower central series shows a Goussarov–Habiro/Stanford "well": the order-2 finite-type invariant a₂ separates k = 2 (value 6) from k = 3 (value 5) and is then frozen for all k ≥ 3 (value 5). Hyperbolicity is graded: the flat twins are torus (B = 8_19, det 3, vol 0; tw_k2 = T(2,7), det 7, vol 0), and the family becomes hyperbolic only at k ≥ 3 (tw_k3 det 67 vol 12.554; tw_k4 det 16323 vol 37.848). Determinant separates every pair (3, 7, 67, 16323), seeing what the finite-type map quotients away. (Status: the well is the Goussarov–Habiro theorem, with the membership hypothesis measured; the torus-to-hyperbolic gradient measured.) Method lesson: v₂ must use a symmetric Alexander normalization; the naive second-derivative form is genus-contaminated and would have falsely reported the well "broken."
F_k = (σ₁³σ₂⁻¹)ᵏ, crossing number 4k: fibered for all k (Stallings, the closure of a homogeneous braid; homogeneity measured and Alexander monic), degree of Alexander exactly 4k − 2, and genus exactly 2k − 1, two-sided pinned — the Alexander lower bound meets the Seifert upper bound, so it is the true Seifert genus (1, 3, 7, 9), not merely a bound. Bonus (not pre-registered): signature exactly −2k. The family is linearly graded: c = 4k, deg Δ = 4k − 2, genus = 2k − 1, signature = −2k; k = 1 is the trefoil bit-identically; k = 3 closes to a 3-component link and is excluded. (Status: fiberedness and exact genus proven; signature law a measured pattern over four points.)
W_k = (σ₁σ₂)ᵏ · σ₂σ₁²σ₂⁻¹, strongly quasipositive: genus exactly k, but not via the naive Seifert bound (the computer-algebra .genus() returns k + 1) — it is pinned via ½·deg Δ = k saturated = the Bennequin/Rudolph band-surface count. Signature is monotone (−2, −4, −6, −8) but obeys no linear law (k = 4 gives −6 ≠ −2k), in sharp contrast to the fiber-forge. Determinant is tiny and non-monotone (7, 9, 3, 5), a weak fingerprint here; k = 1 is 5₂. This family is not fibered (k = 1 Alexander non-monic; strongly quasipositive does not imply fibered). (Status: exact genus proven via the Rudolph/Bennequin pinch; signature measured, monotone only.) Method lesson (general): the computer-algebra .genus() is a diagram Seifert upper bound, never to be read as the true genus; always pinch two-sidedly. Its tightness depends on positivity of the word.
THK(4, n) = (σ₁σ₂⁻¹σ₃)ⁿ, 4 strands: all n ≥ 3 hyperbolic (no torus well), fibered, degree of Alexander exactly 3(n − 1), with a signature ratchet −(n − 1). Two pre-registered bets were falsified and are kept on the record: (i) THK(4,3) ≠ 8_18 (it has determinant 75 and signature −2, where 8_18 has determinant 45 and signature 0; the confusion was THK(4,3) ≠ THK(3,4)); and (ii) the family is chiral, not amphichiral (signature −(n−1) ≠ 0). The contrast with §5.2 is the headline: the 3-strand Turk's-head is amphichiral for every n, the 4-strand loom is chiral — amphichirality flips with the strand number, measured. (Status: fibered/degree measured; the two corrected over-claims are part of the honest record.)
Two Becerra-style ladders show an order-2 finite-type invariant under constructive control: one family dials a₂ in integer steps (0, 1, 2, …, 6, exactly +1 per rung) while keeping signature fixed; a second family freezes a₂ ≡ 0 while determinant and volume explode (to ∼1.1 × 10⁹ and ∼85 respectively). On a two-parameter machine we then tested orthogonality as additive separability: signature and Alexander-degree separate additively (each knob contributes independently), while the quadratic finite-type invariant couples (Goussarov–Habiro polynomiality). Inverting the separable laws gives design-to-specification: solving for the two knobs from a target (signature, Alexander-degree) and building the knot hit the target 3/3 at extrapolation outside the measured grid — genuine predict-then-build, not a fit. (Status: measured patterns; the freeze is the Stanford theorem; the dial and the 3/3 design hits are out-of-sample measured.)
The hard couplings are theorems: amphichiral ⟹ signature 0 ⟹ determinant a sum of two squares (and strongly-amphichiral ⟹ determinant a square); |signature| ≤ 2·genus; deg Δ ≤ 2·genus; determinant odd; signature even. Within those, the (signature, genus) plane for prime hyperbolic knots is reachable (§5.8); adding the determinant axis is solved compositely via connected sum (multiplicative determinant) at the cost of primality, and is an open frontier for prime hyperbolic knots — our first attempt to stack a third twist block was measurably obstructed (every member went non-hyperbolic). A pretzel vehicle reaches all three invariants through separate but coarse knobs (determinant via tangle values, signature via signs in ±2 steps, genus via the count) without collapsing. (Status: couplings proven; obstruction and pretzel reach measured.)
The twisted-torus family T(p,q;r,s) — the (p,q) torus braid followed by s full twists on r adjacent strands — has its determinant fully under control. Base law: det T(p,q;r,s) = det T(p,q) exactly when r is odd and s is even, because the embedded full twist acts as the scalar tʳ·I on the reduced Burau representation, invisible at t = −1 when 2 ∣ rs and 2 ∤ r; this generalizes — the twist is invisible to the reduced Burau at a primitive n-th root of unity iff n ∣ rs and n ∤ r (the determinant law is the case n = 2). For r even the determinant is affine in the twist, det = |a·s + b|, and for p, q both odd the entire family is closed in elementary form: intercept b = det T(p,q) = 1, slope a = 2 Σ_{i<r/2} sgn(p − (q⁻¹(2i+1) mod 2p)) — an explicit signed lattice count — from which the sharp bound |a| ≤ r and the congruence a ≡ r (mod 4) both follow as corollaries (proof, §6.3).
The slope is a Dedekind–Rademacher sign-count and provably not a classical (cotangent) Dedekind sum (a linear sawtooth fit fails for every constant). The p-odd hypothesis is necessary: for p even the reduced Burau is singular (eigenvalue 1 at the j = p/2 mode), the intercept becomes det T(p,q) = q, the apparent non-affineness is only the |·| kink at a sign crossing, and the slope is an irregular Dedekind–Rademacher count with no |a| ≤ r bound. (Status: the determinant law and the p-odd closed determinant are theorems, fully written (§6.3) and machine-checked to 0 violations against genuine SageMath determinants over ∼6000 cases; the p-even slope is now structurally reduced — a nilpotent rank-1 twist makes Δ affine and the slope splits into an elementary part plus a single resolvent term of the singular Coxeter–torus operator, machine-verified, with the write-up honestly scoped in §6.3 and supplement PROOF_peven_slope_structure.md.)
We separate cleanly here what is proved from what is provable but not yet fully written. One result is proven and fully written (§6.3); §6.1 is a known construction (Conant–Manathunga 2016) whose proof we write out and check; one is a correct obstruction with named gaps (§6.2).
Construction. Let A be a braid word and let m be the operation that flips the sign of every generator (the mirror). Form W = (A · m(A))ⁿ and take the standard (trace) closure K = closure(W), oriented by the braid direction. The only hypothesis is that the second half of each factor is the exact mirror of the first (B = m(A)); the syllable count 2n is then automatically even. (Correction v0.25: through v0.24 this read "trace (plat) closure" — the plat closure is a different operation, and the proof uses the standard closure — and listed the even syllable count as a separate precondition.)
Theorem. K is positive amphichiral: it is isotopic to its mirror image as an oriented knot, i.e. some orientation-reversing homeomorphism of S³ carries K to itself preserving the orientation of K. In particular K is amphichiral as an unoriented knot.
Proof. Reflection in the projection plane is an orientation-reversing homeomorphism of S³; it carries the closed braid of W to the closed braid of m(W) and keeps the direction of every strand, so closure(m(W)), oriented by the braid direction, is the oriented mirror image of K. Since m respects concatenation and is an involution, m(W) = (m(A) · A)ⁿ = A⁻¹ · (A · m(A))ⁿ · A = A⁻¹ W A — a cyclic permutation of the syllables of W, i.e. a conjugate braid. Conjugate braids have isotopic standard closures as oriented links (Markov conjugation / the trace property). Hence K is isotopic to its mirror image as an oriented knot. ∎
What this does and does not say. It gives the positive class: the mirror symmetry preserves the orientation of K. Whether K is also negative amphichiral — for a positive amphichiral knot this is the same as K being invertible — is not decided by the argument, and in general it fails: for A = σ₁⁻¹σ₂³σ₁⁴, n = 2 (hyperbolic, volume 40.22) SnapPy finds an isometry of the complement to its mirror that fixes the peripheral framing (the positive witness) but none that negates it (the negative witness); the reading of the cusp maps was calibrated on 4₁ (both classes), 8₁₇ (negative only) and 10₇₁ (chiral), SnapPy 3.3.2, 2026-10-25. (Correction v0.25: through v0.24 this paragraph said the oriented class was not settled; it is, and the upgrade was pointed out by another AI system reading the public text, 2026-10-24.) The "Shubnikov / point-group color-swap" language that motivated the construction is motivation: the proof lives entirely in the braid group and makes no claim that the realized space curve exhibits a measured color-swap symmetry. The one hypothesis (exact mirror halves) is stated; nothing else is needed. The result has now been corroborated through the invariant pipeline on five explicit instances (A ∈ {σ₁σ₂, σ₁σ₂σ₁σ₂⁻¹-type, σ₁σ₂σ₃σ₄, …}): in every case the proof mechanism holds directly (m(W) is a cyclic permutation of W), the necessary invariants agree (signature 0, and the Jones polynomial of K equals that of its mirror), and — the sufficient condition — the hyperbolic complement is isometric to its mirror (SnapPy, volumes bit-matching to their mirror).
The examples recover known amphichirals (4₁ at volume 2.0299; 8₁₈ at 12.3509) and even reproduce a corpus anchor (volume 13.417374 = 12n_706, the celtic-for-playing knot), alongside larger instances (det 117, 3509, 14157). Prior art — this construction is known, and we claim no novelty for it (checked 2026-08-10). Conant and Manathunga state exactly this construction, without proof, as a standard systematic method: "Let w be a (2n + 1)-braid, and w\ be the mirror image of the braid w (which means all crossings are reversed). Then the closure ww\ is a strong positive amphicheiral knot and the closure ww\ww\ is a periodically positive amphicheiral knot" [CM16, §3, with Figure 3].
Their conclusion is finer than ours: "strongly" and "periodically" positive amphicheiral mean that the symmetry is realised by an involution, resp. by a map of finite order, which our argument does not provide; our hypothesis is slightly broader (any strand number, any repetition count). Independently, Gittings gives an explicit braid-word symmetry as a sufficient condition for amphicheirality — "If a braid that has been rotated is equal to the reverse of the original braid, then the corresponding knot or orientated link is amphicheiral. This is a sufficient condition, but not a necessary one" [Git04, §5] — a different symmetry from ours, but the same idea and twenty years old. The two ingredients of our proof (the mirror of a closed braid is the closure of the sign-flipped braid; conjugate braids have the same closure) are textbook braid theory [Bir74].
Finally, the classical special case A = σ₁, giving the 3-strand Turk's head (σ₁σ₂⁻¹)ⁿ, already appears two sections earlier in this report (§5.2) and is long known to be amphichiral. What remains ours is only the write-up: a four-line self-contained proof of positive amphichirality for arbitrary strand number and arbitrary repetition count, without the "strongly/periodically" refinement. That is a remark, not a result, and it is presented here as such. Full receipts, including how each source was obtained and read, are in paper/PRIOR_ART_GATE_6_1_20260810.md.
Claim. A smooth embedded closed space curve cannot carry a Cₙₕ point-group symmetry for n ≥ 3. (Empirically: 200/200 attempts to construct such a curve in our family produced a self-intersection — minimum non-adjacent distance exactly 0 — so the embedding gate rejected every one.)
The obstruction argument. A Cₙₕ symmetry includes the horizontal mirror σ_h. Acting on the parameter circle S¹, the orientation-preserving case of σ_h is conjugate to the half-turn t ↦ t + π; this forces the planar coordinates to satisfy xy(t + π) = xy(t) and the height to satisfy z(t + π) = −z(t). A continuous z with zero mean has a zero t\; since the half-turn is fixed-point-free, P(t\) = P(t\* + π) is a self-intersection. Hence no embedding. The orientation-reversing case is handled separately.
Why this is not yet a theorem (gaps named honestly).
Disposition for the paper. §6.1 is a known construction (prior art above), written out with proof and checks; the proven result of this section is §6.3. §6.2 is presented as a result-in-progress with a correct obstruction and explicit gaps — included because honesty about the proof state is the point, and because completing gaps (1)–(2) is a well-defined, finite task. We expect to either complete it or downgrade it to a measured construction-family fact in a later version, and we mark which.
Construction. For p, q odd with gcd(p,q) = 1, r even, 2 ≤ r < p, and a twist count s ≥ 0, let T(p,q;r,s) be the closure of the p-braid (σ₁⋯σ_{p−1})ᵠ · (σ₁⋯σ_{r−1})ʳˢ. Write C for the reduced Burau matrix of the Coxeter element δ = σ₁⋯σ_{p−1} at t = −1, M_T = Cᵠ, and u, v for the twist covectors (u the indicator of the last p − r coordinates; vᵢ = (−1)ⁱ on the first r − 1).
Theorem. The Alexander determinant is
det T(p,q;r,s) = | 1 + s·a |, with a = 2 Σ_{i=0}^{r/2−1} sgn( p − (q⁻¹(2i+1) mod 2p) ).
In particular det is affine in the twist with intercept 1, the slope satisfies the sharp bound |a| ≤ r, and a ≡ r (mod 4).
Proof (outline; full write-up with every identity symbolically verified in the supplement PROOF_slope_bound_a_le_r.md). (i) The reduced Burau of δ at t = −1 is the companion matrix C = 𝟙bᵀ + S (𝟙 all-ones, S the subdiagonal shift, b_j = 2(−1)ʲ with b_{p−2} = −1), proved by induction on the partial products δ_k = σ₁⋯σ_k via the column-update rule of right-multiplication by σ_k. (ii) Since p is odd, the eigenvalues −ζ_pʲ give Cᵖ = −I, hence (M_T − I)⁻¹ = −½ Σ_{k=0}^{p−1} M_Tᵏ, and a cofactor expansion gives a = −det(M_T − I)·Σ_{k} t_k with t_k = vᵀM_Tᵏu. (iii) A generating-function computation on the orbit recurrence yields a closed form g(z) = Σ g(m)zᵐ = −z(1−zʳ)(1+z^{p−r})/((1−z²)(1+zᵖ)) for g(m) = vᵀCᵐu, whence g ∈ {−1,0,1} (Lemma A) and t_k = g(qk mod 2p). (iv) The support of g is invariant under m ↦ m + p mod 2p; with q⁻¹ a bijection of ℤ/2p and q(k+p) ≡ qk + p, exactly r of the t_k are nonzero (Lemma B), and a head–tail symmetry m ↦ p − m collapses Σ t_k to the signed lattice count. (v) det(M_T − I) = 1, being the determinant of the base torus knot T(p,q) = 1 for p, q odd (an elementary cyclotomic product). The bound and congruence are immediate corollaries (r/2 signs of ±1; the count is even). ∎
Scope (the p-odd hypothesis is necessary) and the p-even structure. For p even the eigenvalue −ζ_p^{p/2} = 1 makes C singular (Cᵖ = +I, det(M_T − I) = 0), so the p-odd closed form does not apply; the determinant is still affine, det = |q + a·s|, and the slope is an irregular Dedekind–Rademacher count, provably not a classical (cotangent) Dedekind sum. This slope, once an unstructured open frontier, is now structurally reduced (supplement PROOF_peven_slope_structure.md): the embedded full twist Δ_r² = (σ₁⋯σ_{r−1})ʳ acts at t = −1 as a nilpotent rank-1 update W = I + a bᵀ (disjoint supports, so bᵀa = 0 and Wˢ = I + s·a bᵀ), whence A_s := M_T Wˢ − I = A₀ + s·(M_T a)bᵀ is exactly linear in s, the torus left-null vector v is a common left-null of all A_s (vᵀa = 0), and Δ(s) is provably affine. The whole slope then decomposes as
a_even = (2/p)·[ κ·((r/2)·vᵀT′a + vᵀU′w₀) + vᵀT′m ],
an elementary part plus a single resolvent term vᵀT′m, where m is the regularized inverse of the singular Coxeter–torus operator A₀ against the twist vector — the sole non-elementary ingredient, a classical Dedekind–Rademacher lattice sum. So the p-even case is closed as far as linear algebra reaches, with the difficulty precisely localized (contrast p odd, where M_T − I is invertible and no resolvent obstruction arises). (Status: p-odd determinant — Theorem, fully written, ∼6000 cases 0 violations. p-even structure — the structural reduction is now proven: the twist is a nilpotent rank-1 update (rank exactly 1 via a left/right-invariance argument, L2a/L2b), Δ is affine (the s²-coefficient vanishes because vᵀU′ ∝ bᵀ from the universal covector and bᵀm = 0 by Sherman–Morrison), and the slope decomposes as elementary + resolvent (14/14 exact). The only open ingredient is the elementary closed form of the single resolvent term vᵀT′m — a genuine cotangent/Dedekind–Rademacher sum over the torus spectrum {−ζ_pʲᵠ}, honestly open. Supplement PROOF_peven_slope_structure.md.)
Mutation preserves the entire classical stack we compute. We verified this on our own machine-built Conway/Kinoshita–Terasaka pair (crossing number 11): Alexander = 1, determinant = 1, Jones bit-identical, signature 0, hyperbolic volume identical to 13 places, and θ bit-identical on this pair (θ does separate other mutants, §2.5). Every cheap escape is dead by construction: cable volume (a cable is a satellite, hence toroidal, hence non-hyperbolic), the cable Alexander (companion Δ = 1), the symbolic cable HOMFLY (a 3-cable of an 11-crossing knot is ∼99 crossings and the symbolic computation dies), and all-color colored Jones (mutation-invariant).
We separate the pairs with a self-built numerical Hecke–HOMFLY (a 3-cable). The pipeline is: q-seminormal Hecke representations (relation residual ∼10⁻¹⁶) → Ocneanu weights, calibrated against the computer algebra system on the trefoil and predicted on the figure-eight → framing correction → assembled HOMFLY; the underlying Reshetikhin–Turaev evaluator validates bit-exactly against the computer-algebra HOMFLY on the unknot, trefoil, and figure-eight.
Results. For the Kinoshita–Terasaka/Conway pair at one evaluation point, the 3-cable values differ (3031.553 vs 3030.473, |Δ| = 1.08) while the 2-cable values are equal (|Δ| ∼ 10⁻¹⁴) — the latter both confirming Morton's result that the 2-cable does not separate and demonstrating that the method detects equality to machine precision. A second evaluation point gives |Δ| = 33.2 (3-cable) with the 2-cable again equal. The same procedure separates 13a₁₂₃₁/13a₁₂₃₇ (|Δ| = 27.7 at the 3-cable; 2-cable equal). Since the difference of HOMFLY polynomials is a Laurent polynomial, a single non-zero value proves non-vanishing, hence separation; two points, two pairs, and consistent 2-cable equality make it robust. The raw magnitudes are point-dependent witnesses, not invariants; the provable claim is that the polynomials differ.
Honest attribution. The mathematics is Morton's (1996) Hecke-trace HOMFLY. Our contribution is the scalable computation: the standard library times out on the 12-strand cable, where the Hecke-trace computation completes in about 13 minutes per knot. This is the same asymmetry as the rest of the project — a robust, possibly inelegant computation that runs where the elegant one stops.
Data availability. Every knot in the corpus has a public record: the centerline curve, the full invariant fingerprint, a diagram (a planar-diagram SVG where a minimal diagram is legible, otherwise a braid diagram), and a rotatable 3D viewer. The site knot-structures.stainlesssteel4u.de carries all sculptures and constructive clusters (it is fully static and public). The corpus database is the single source of truth for all counts and identities reported here.
The working protocol — reproducibility lives in the process, not the model. The language model is non-deterministic and retains nothing between sessions; the process is what is deterministic. The collaboration runs a fixed per-session discipline: a boot sequence (load identity, a curated memory index, and a living "NOW" working-thread stating what is running / waiting / next); a single source of truth with artifacts written durably from creation; an append-only discoveries-and-error log that records each failure and the rule it forced; and a mandatory close that rebuilds the state mirrors and verifies each output file at the byte level — a loud failure instead of a cheerful "done" over an empty file. After a full restart the working context is reconstructed entirely from these artifacts. This is why a months-long program across hundreds of sessions stays coherent despite a stateless collaborator.
Drift is caught, not prevented. We do not claim the model does not err, forget, or over-conclude — it does, within and across sessions. The claim is narrower and is the honest one: the protocol makes such drift cheaply detected and recovered, and converts each incident into a durable rule. The corrections recorded in §9 — the Frankenstein audit, the resolution caveat, the over-claims that measurement later falsified — are not embarrassments hidden after the fact; they are the protocol working. A recurring operational failure is promoted from the archive into a boot-visible checklist precisely because a lesson that lives only in the archive does not prevent its own recurrence.
Relation to concurrent work on agent hygiene. We claim no novelty for the bookkeeping mechanisms as such. A concurrent and independent system, Ratchet (2026), formalizes nearly the same hygiene recipe — an append-only evidence log, never-delete retirement, failure-cluster-driven curation, "the bottleneck is the librarian, not the author" — for autonomous agents, with a formal non-divergence guarantee and benchmark validation that we do not have. Our setting is the inverse: a hand-authored protocol for human–AI collaboration, validated by real mathematical deliverables rather than benchmarks. The honest relationship is convergent validation, not priority.
Where this report is published. Two complementary venues:
Further deposits not discussed in this report. §4 lists our contributions to problems posed by others; the following deposits are not discussed here: Theta on constructed Conway-mutant pairs: thirty separated, one not (DOI 10.5281/zenodo.21946255; see §2.5), Lower bounds for the unknotting number over a corpus of built curves (DOI 10.5281/zenodo.21933898), Congruences on the Conway polynomial forced by Conant's Corollary 7.3: an answer to the first half of Question 9.2 (DOI 10.5281/zenodo.21596227), 315 positive amphicheiral hyperbolic knots whose Conway polynomial does not split (priority deposit, DOI 10.5281/zenodo.21590760), and The leading coefficient of the Alexander polynomial of twisted torus knots (DOI 10.5281/zenodo.22769534, cited in C10). The preprint shelf of the website lists every deposit with its abstract. All DOIs in this report are concept DOIs (they resolve to the newest version), except where a version is named.
Reproducibility boundary. The invariant computations, the corpus, and the anchor recomputations are fully reproducible from public data and standard tools. The kinematic resolution mechanism (how crossings are resolved over the camera sequence) is deliberately not disclosed; nothing in the claims depends on trusting that mechanism, because every identity is cross-checked by independent standard invariants and, where in range, against tables.
We treat our own corrections as part of the method.
double-6 from anchor A2 are outcomes of that audit.spherogram, with reducedness established from the absence of articulation points in the underlying 4-valent projection graph), or span_t V(K) equal to the diagram's crossing count. 341 of our 843 records carry such a certificate (206 of the 504 distinct knots, 91 of those records non-alternating; counted 2026-09-09, distinct count updated 2026-09-19 — the pair merged in v0.10 carried no exactness certificate (span 18 against an upper bound of 24), so 206 is unchanged); the remaining ones keep the bound. Since v0.8 the genus–braid-index bound of §3.3 supplies a further certificate wherever it reaches the diagram's crossing count — for four of the seven knots promoted in v0.8 — and the count above has not yet been re-measured against it. The distinction is machine-checked per record, not asserted globally.slice_verdict.py searches obstructions (Fox–Milnor/determinant, σ, Arf, τ, s). All doors open means "undetermined", never "slice" — and that is what it reports. Positive slice verdicts therefore enter only through a small, hand-maintained proof register, and only with a written proof or a named source attached; the register is deliberately readable so it cannot become a back door for claims. Two entries are literature receipts (A7), two are our own arguments (A1). A first piece of the constructive side now exists — a gate (symmetrische_vereinigung.py) that tests a record against a claimed symmetric union via det = det(K±)², with the partial-knot determinant computed rather than looked up, and with rejection cases built in before the builder; T(2,9) passes the determinant test and is stopped only by σ, which is why that check is not redundant. We still lack the builder itself — a band presentation the pipeline builds and verifies itself. That is the honest gap: we can currently recognize a ribbon knot only if someone else has drawn the band.W.S. conceived the sculptures and the constructive aesthetic, drew the centerline of every sculpture, provided the hardware, posed the questions that drove the program, decided its direction, at times proposed how a problem could be attacked, and demanded that every claim be checked before it stands. C.C. wrote the constructive generators (the source of the machine-family centerlines) and the code that orchestrates established research software — SnapPy, spherogram, Regina, SageMath, KnotJob and Bar-Natan–van der Veen's Theta.sage, among others — into the identification pipeline; ran and audited the corpus; led the proofs and proof sketches of §6, with W.S. as coauthor there; and drafted this report. No invariant value in this work comes from the language model: every one is computed by those tools (see the coauthorship statement). Both authors take responsibility for the integrity of the claims as labeled. The first author is a sculptor, not a credentialed mathematician; the mathematical reasoning is native rather than institutional, and the labeling discipline of §1.3 is how we keep that reasoning honest.
Dates: v0.26: 2026-11-06; v0.25: 2026-10-25; v0.24: 2026-10-17; v0.23: 2026-10-16; v0.22: 2026-10-08; v0.21: 2026-10-08; v0.20: 2026-10-06; v0.19: 2026-10-05; v0.18: 2026-10-03; v0.17: 2026-10-03; v0.16: 2026-09-26; v0.15: 2026-09-26; v0.14: 2026-09-26; v0.13: 2026-09-25; v0.12: 2026-09-25; v0.11: 2026-09-20; v0.10: 2026-09-19; v0.9: 2026-09-18; v0.8: 2026-09-11; v0.7: 2026-09-09; v0.6: 2026-09-06; v0.5: 2026-09-05; v0.4: 2026-08-10 census refresh; v0.3: 2026-07-15 external contributions C1–C3 in §4; v0.2: 2026-06-30 methodology §8; v0.1: 2026-06-19.
labeled_at 2026-09-04) and sharpens the novelty boundary (§3.3): prime knots are enumerated through c = 20 (Thistlethwaite 2025), so "novel" now requires a proven lower bound c ≥ 21 — via the Jones-span bound (Kauffman–Murasugi–Thistlethwaite) or via c ≥ 2g+1 from the HFK-exact Seifert genus. Seventeen former candidates were retracted against Burton's c ≤ 19 census — among them the sculpture knot trainstation/sundown (= 18ah_5090151) — and 82 unmatched knots moved to an explicit "unresolved" class inside the enumerated range.is_isometric_to, calibrated against the same knot from a second projection) separates them, so they are counted as two. Counts move to 843 records / 504 distinct / 128 known / 287 novel-hyperbolic / 47 unresolved, headline 220 (unchanged: both knots are tabulated).poly_s08 and kf332_polys08 agree in determinant (25721), signature (0), Jones polynomial and HOMFLY polynomial bit-for-bit, Seifert genus (9), fiberedness, θ (identically zero, both amphichiral) and hyperbolic volume to fourteen decimals, and their complements are isometric. They were separated only because the Alexander polynomial — defined merely up to ±t^k — was stored once as −Δ and once as +Δ, and our distinctness key compared the string. The key now compares a normal form (leading coefficient positive; the t^k shift is deliberately not normalized, as no such case occurs in the corpus). A sweep over all distinct representatives finds exactly this one pair. Counts move to 843 records / 503 distinct / 128 known / 286 novel-hyperbolic / 47 unresolved, headline 219.celtic-for-playing and confirmed independently by A. Zupan; with them, the set R^s_5 (145 Alexander polynomials). Census counts are unchanged.nitro (A4) lies in the unresolved class c ≥ 20, not in novelty; C1 is withdrawn as a completed verification, because a validity check found wrong Kauffman a-spreads in the stored tables, and its c = 15 count is completed to 253,293; the 388 "novelty singletons" are records in a folder, not novel knots; the abstract counts three sculpture↔machine identifications, not four; §1.2 names both alternating boundaries; the coauthorship statement is brought in line with §6.2 and §10; the A7 cross-reference is fixed; the version strings agree. v0.23 also corrects the claim that θ is mutation-invariant (§2.5, §7, §9; our own deposit of 2026-08-15 shows θ separating thirty mutant pairs), lists the deposits not discussed here (§8), cites C1 by its concept DOI, and explains why papers and report name their authors differently. It adds C12, our first arXiv paper (arXiv:2610.16600), and brings §8 up to date with the arXiv endorsement and arXiv's submission limit.Placeholder — to fill with full citations before submission.
FAMEDexploration (A. Guilloux). T. Liu, S. Ming, X. Sun, B. Wu, T. Yang, U_qq̃ sl(2; ℝ) Turaev–Viro invariants for cusped 3-manifolds, arXiv:2608.16560 (2026) — C5.khovanov_homology, khovanov_polynomial, khovanov_width, khovanov_torsion, khovanov_thin) and the Rasmussen s-invariant were computed with it, across 726 records (counted 2026-08-15). The pipeline calls KnotJob.jar directly; none of that arithmetic is ours.Draft v0.25. Evidence-status tags (theorem / measured pattern / conjecture) are load-bearing and are not to be softened in editing. All counts trace to the live corpus database (labeled_at 2026-09-20). All anchor values are bit-verifiable from public data. §6.1 is proven, but it is not ours: the prior-art review of 2026-08-10 found the construction published (Conant–Manathunga 2016, with a stronger conclusion; see §6.1 and paper/PRIOR_ART_GATE_6_1_20260810.md). It stands here as a correct proof of a known fact and must never be presented as an original result. §6.2 is a result-in-progress with named gaps. The working-protocol material in §8 is method, not a mathematical claim. — W.S. & C.C., 2026-10-16.