Draft v0.2 — 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 v0.8 — 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). v0.5 refreshes every census count against the live corpus database (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. v0.6 completes the census comparison through c = 19, in every layer: the 17-19 stream covered all 350 449 955 published census knots (every category matching its source count exactly, zero determinant failures), a retriangulation pass rescued 35 445 of 35 477 failed census volumes, and — after our own consistency guard caught a knot with a 15-crossing diagram being declared "absent" — the c ≤ 16 layer was closed rigorously as well (all 1 701 903 hyperbolic census knots at 3-16 crossings streamed, category totals matching the known prime-knot counts 14/14, zero failures). Outcome: thirteen further candidates retracted (30 in total; three of them, at 15-16 crossings, had slipped through the earlier lookup layer — every retraction isometry-confirmed, census name on record), and — after a final high-precision pass (quad-double arithmetic, up to 40 retriangulations per manifold) rescued the last eleven census volumes that had resisted verification — all unresolved knots except one are proven c ≥ 20 by rigorous absence from the complete c ≤ 19 census, among them the sculpture knots Sundowner, Nitro and Cosmic-Love (the single holdout is a c ≤ 595 giant whose determinant is not yet computed). Closing the census layers also completed the missing knot-Floer data: two unresolved knots turned out to have HFK-exact Seifert genus 13, hence c ≥ 2g+1 = 27, and were promoted to novel. The built-in counter-check held at every layer: none of 25 span-certified c ≥ 21 knots appears anywhere in the census. Counts move to 127 known / 54 unresolved (53 of them with proven c ≥ 20) / 279 novel-hyperbolic, headline 212. v0.7 closes that last gap: the holdout’s determinant is now computed — 953524026999049021263 (21 digits), two independent routes (pointwise Alexander over finite fields with CRT reconstruction, and a 1166×1166 Seifert matrix via fraction-free Bareiss elimination) agreeing bit-for-bit — and a determinant-level census sweep (every census knot’s determinant computed exactly from its Regina signature; all six 17–19 category totals matching their source counts exactly, zero failures; the 3–16 layer likewise complete) shows this determinant appears nowhere in the complete c ≤ 19 census: the holdout is proven c ≥ 20, so all 54 unresolved knots now carry a rigorous c ≥ 20 lower bound. The same sweep proved two further determinants absent, completing the salvage of the pipeline-failure record kf320 (→ novel-hyperbolic, span-certified c ≥ 32) and covering one measured-only record that remains outside the corpus. Counts move to 843 records / 503 distinct / 280 novel-hyperbolic, headline 213. v0.8 extends the genus route of the lower bound (§3.3) from c ≥ 2g+1 to c ≥ 2g + b − 1, with b a proven lower bound on the braid index — the Morton–Franks–Williams bound from the HOMFLY polynomial where available, otherwise b ≥ 2 for a knot proven non-trivial — and with the genus entering as the HFK-exact value or, failing that, as g ≥ ½·span Δ. Seven unresolved knots now carry a proven c ≥ 21 and move to novel-hyperbolic: four of them with the crossing number determined exactly (lower bound equal to the diagram's crossing count; three at c = 24, one at c = 21), one at c ≥ 22, one at c ≥ 21, and one c ≤ 262 giant at c ≥ 65 (genus from the Alexander span, braid-index floor b ≥ 2); no distinct knot's lower bound decreased. Counts move to 843 records / 503 distinct / 287 novel-hyperbolic / 47 unresolved (every one of them with proven c ≥ 20), headline 220. 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 physical steel sculpture (Werner Stanggassinger); the questions, the hardware, and the artistic family of constructions are his. The software — the pipeline and the constructive generators, the corpus management, and the prose — is an Anthropic large language model ("Claude Code"). A boundary we hold throughout, and which is load-bearing for trust: the language model has no mathematical authority. It writes and orchestrates software; it does not compute the invariants and does not stand in for proof. Every invariant is computed by established tools (SageMath, SnapPy, Regina, KnotJob) and independently re-verified; every proof here is elementary and fully written out, with its identities checked symbolically. 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. 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: (i) a census of 503 distinct knots across 843 realizations, of which 287 are novel hyperbolic candidates, each carrying a proven lower bound c ≥ 21 (Jones span, Kauffman–Murasugi–Thistlethwaite, or the genus–braid-index bound c ≥ 2g + b − 1) and hence lying beyond the complete enumeration of prime knots through c = 20, while 47 further unmatched hyperbolic knots are held in an explicit unresolved class inside the enumerated range (mirror-collapsed counts; live source of truth: the project corpus database); (ii) a set of bit-exact verifiable anchors that any reader can recompute, including a six-fold connected sum of trefoils, a torus knot T(3,7), and four sculpture↔machine identifications; (iii) a collection of measured invariant laws for the constructive families (determinant multiplicativity, signature/genus additivity, fibered families with exact genus, an amphichirality that flips with strand number, a constructively dialable finite-type invariant, and design-to-specification of two invariants simultaneously); (iv) proven results, with their provenance stated — an unoriented amphichirality of a braid-word construction, for which we give a short self-contained proof but claim no novelty (the construction is published, Conant–Manathunga 2016, with a stronger conclusion; §6.1), and a closed-form Alexander determinant for the twisted-torus family T(p,q;r,s) with p, q odd, det = |1 + s·a|, whose explicit signed-lattice-count slope yields the sharp bound |a| ≤ r and the congruence a ≡ r (mod 4) as corollaries — together with a third result (impossibility of Cₙₕ symmetry for an embedded closed curve, n ≥ 3) for which we give the obstruction proof and state explicitly where the write-up is still a sketch; and (v) a scalable numerical Hecke–HOMFLY computation that separates mutant pairs (Kinoshita–Terasaka/Conway and 13a₁₂₃₁/13a₁₂₃₇) where the classical invariants and the Bar-Natan/van der Veen θ invariant are blind. 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 — and the remaining fourteen are 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 is an experimental-mathematics report, in the tradition of computational discovery followed by honest labeling of evidence status. It is deliberately not a proof-paper. Most of our structural findings are measured patterns — verified by computation across a family, but not proven. 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.
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 mutation-invariant and therefore blind on mutant pairs. We verified this directly: on the Conway/Kinoshita–Terasaka pair and on 13a₁₂₃₁/13a₁₂₃₇, θ is bit-identical. This is not a defect we discovered late; it is a known property of θ, and it is exactly the constraint that motivates the separate mutant-separation computation of §7. We flag it as an outreach question (the resolving power of θ on the mutation axis) rather than as a result.
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 2026-09-11), which is the single source of truth; any number hardcoded elsewhere in the project is treated as stale.
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).S_24_d81_D6_amphi (26 members, the ping-pong anchor, A5) and S_18_d243_D3_chiral (24 members).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 ≤ 16 (the snappy HTW census); 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. Prime knots are enumerated through c = 20 (Thistlethwaite, Algebr. Geom. Topol. 25 (2025) 329–344: 1 847 319 428 knots, independently confirmed by Burton), but the c = 20 data are not public: a knot below that line 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) its crossing number is proven ≥ 21 by 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.) All 287 novel-hyperbolic distinct knots carry such a certificate: they lie beyond every existing enumeration. The 47 unmatched hyperbolic knots without one form an explicit unresolved class — they sit in a fully enumerated range, and their unmatched status is a statement about our tables, not about the knots. Even for the 287, "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 42 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, no census/Fremlin match — a novelty candidate, and the first sculpture↔machine match inside genuine novelty (unlike A2, which points into the table). 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 and mirror — both our own K # m(K) arguments (A1) — 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. (Caveat carried forward: Lamm names 12a435 as undecided, so the list of seven may grow. Any external use of this statement must re-check it.)
External contributions (first, dated, verifiable). Three results contributed to open problems posed by others, each recomputable from public data and deposited for a citable, non-editable timestamp on Zenodo — DOI 10.5281/zenodo.21389681 (https://doi.org/10.5281/zenodo.21389681, published 2026-07-16, CC-BY-4.0).
C1 — Jabłonowski's c7 inequality, verified beyond 13 crossings. The conjecture sp_a(P) ≤ sp_a(F) (the a-spread of the HOMFLY-PT polynomial is at most that of the Kauffman polynomial), stated open in M. Jabłonowski, Integer Knot Invariants: Inequalities, Computations, and Open Problems (arXiv:2605.22652, 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 verified it for all 46,972 knots at c = 14 (minimum slack 2) and all 253,281 knots at c = 15 (minimum slack 4), with zero violations — to our knowledge the first verification at 14 and 15 crossings, as KnotInfo does not tabulate the 14-crossing Kauffman a-spread. Reported to the author 2026-07-09; positively acknowledged 2026-07-11. Columns of the deposited tables: knot, spF_a, spP_a, slack, c7_holds.
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.)
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 → novel, hyperbolic, det 841, 2205, 15125, 39601, volumes to 43.12. (Status: amphichirality proven per instance via complement isometry; the determinant-form law is a measured pattern; novelty of n ≥ 7 measured.)
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. Two results are proven and fully written (§6.1, §6.3); 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 trace (plat) closure K = closure(W), with the standing preconditions that the syllable count is even and B = m(A) (so the two halves are exact mirrors).
Theorem. K is amphichiral as an unoriented knot.
Proof. Mirroring the closed braid corresponds to applying m to the whole word; one checks that m(W) is a cyclic permutation of the syllables of W. Trace closure is invariant under cyclic permutation of the braid word (Markov conjugation / the trace property). Therefore closure(m(W)) = closure(W) as unoriented knots, i.e. mirror(K) = K. ∎
What this does and does not say. It licenses unoriented amphichirality only; the oriented ± class is not settled by this argument. 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. Preconditions (even syllable count, exact mirror halves) are necessary and are stated. 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 stronger than ours (strongly/periodically positive amphicheiral — an oriented statement); 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 the unoriented case for arbitrary strand number and arbitrary repetition count. 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 the backbone proven result. §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 mutation-invariant. 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:
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 (205 of the 503 distinct knots, 91 of those records non-alternating; counted 2026-09-09); 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, provided every centerline geometry and the hardware, and posed the questions that drove the program. C.C. wrote the pipeline and the constructive generators, ran and audited the corpus, performed the invariant computations and the proofs/sketches above, and wrote this report. 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.
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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 v0.8. 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 2026-09-11). 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., 2026-09-11.