Every object here, whether a stainless-steel sculpture or a machine-generated construction, enters the pipeline as one thing only: its centerline curve, stored as a Fourier series. Everything below is derived from that curve.
From the centerline alone the pipeline (1) scans the curve on the GPU and projects it to locate its crossings; (2) tracks the crossing pattern into a knot diagram; and (3) computes a complete set of invariants directly from that diagram — Alexander, Jones and HOMFLY polynomials, determinant, signature, Seifert genus, hyperbolic volume and symmetry group, and a canonical Theta-cluster hash. The output is the knot’s identity — or the verdict that it lies outside every catalogue, as a genuinely new object. All 510 centerlines here went through exactly this path.
The computation rests on established, open tools, and every one of them is somebody's work: a CUDA scanner for the curve (ours), regina (Burton, Budney, Pettersson and others) and SnapPy (Culler, Dunfield, Goerner, Weeks) for diagram and hyperbolic invariants, spherogram (part of the SnapPy project) for diagram handling, sympy for the polynomial algebra, KnotJob (Dirk Schütz, GPLv3) for Khovanov homology and the Rasmussen s-invariant, and the Theta invariant of Bar-Natan and van der Veen, computed with their own Theta.sage. The reference catalogues — KnotInfo (Livingston–Moore), the SnapPy census, and a numerical Theta-DB across the 59,937 knots.sage entries — are used only as described next.
We name the authors because the invariants on this site are not all our own arithmetic: the Khovanov columns come out of Dirk Schütz's KnotJob, the Theta column out of Bar-Natan and van der Veen's Theta.sage. What is ours is the pipeline around them — curve to diagram to identity — and the corpus it produced.
The pipeline runs blind: in the primary identification path there are no table lookups. Every invariant is computed from the centerline itself. The catalogues (KnotInfo, SnapPy census, Theta-DB) form a separate cross-check layer that runs only after the invariants are already computed.
So an identification is never circular — the machine never “looks up the answer.” It computes the full fingerprint from the geometry, and only then asks whether any catalogue contains that fingerprint. When the answer is “no catalogue entry,” the novelty is a measured result, not an assumption; and when a sculpture and an independent machine construction land on the same fingerprint, that agreement was reached separately on each side.
To check the machine against answers it could not know, we took knots from a public knot catalogue, stripped their identity, gave them fantasy names, rebooted to a clean state, and ran them blind. Only at the end of the session did we compare the verdict against the catalogue’s known classification:
That is the whole point of running blind: a correct identification is earned from the geometry, not retrieved from a table.
The identification and the blind validation ran on a fixed, version-pinned environment, recorded in a machine snapshot so that the results are reproducible and free of silent tool drift; the no-lookup discipline is enforced in code (a pipeline seal blocks catalogue lookups in the primary path). The setup: an AMD Ryzen 9 3900X (12 cores), 32 GB RAM and an NVIDIA RTX 3070 (CUDA 13) under Linux 6.17, running SnapPy 3.3.2, regina, SageMath 10.7, NumPy 2.4, SymPy, CuPy and the Theta.sage tool. Full machine & version description ↓
The discipline that runs this project — a fixed boot order, a working diary, growing files of numbered lessons and raw failures, and gates that must prove they can say no before their yes counts — exists as a small, self-contained folder: Markdown and shell scripts, no vendor, no key. You place it in your own environment and start the AI tool of your choice inside it. Its one load-bearing rule: at first, the AI only observes and writes — it changes nothing (we recommend three days). A measuring script — not a calendar — decides when that phase is done, and it can say no. That rule is a recommendation, not a lock: the folder is yours from the first minute.
workshop-kit.zip ↓ (sha256) — a gift, licensed CC BY 4.0: use, modify, redistribute, commercially too; attribution is the only condition. It needs nothing but a shell. We run it on Linux, and that is the only platform it has actually been run on — macOS is source-reviewed but has never been started, Windows needs WSL or Git Bash. Best results on a spare Linux machine. Folder, files and scripts are entirely in English. Honestly: both trial runs so far were in our own environment — outside our workshop the folder has not yet been used. And read-only access stops changes, not data leaving the house: with an AI that runs in a data centre, whatever it reads goes to its provider — clear that first.
On the page of every hyperbolic knot there is a ♪ play tone button. The pitch you hear is not decoration and is not chosen by us — it is read straight off the knot’s own geometry.
A hyperbolic knot’s complement — the space around the strand — carries a rigid hyperbolic geometry, and inside it sit closed loops of shortest possible length, its geodesics. The collection of their lengths is the length spectrum, a genuine invariant: it depends only on the knot, never on how the curve happens to be drawn, scaled or parametrized. We turn each length into a frequency in direct proportion — a fixed number of hertz per unit of length — so the shortest geodesic, the systole, becomes the fundamental, and the longer geodesics become its overtones. The scale is absolute, with no per-knot normalization, so two different knots genuinely sound at different pitches and the same knot always sounds the same. What you hear is the knot’s intrinsic chord.
Two honest limits come with it. Because the length spectrum is mirror-symmetric, a knot and its mirror image sound identical — the ear alone cannot hear handedness, though the Theta hash can. And not every knot has a tone of this kind: torus knots and connected sums have no hyperbolic structure, hence no length spectrum — that silence is itself a true statement about the knot. Where only the curve itself is available, the page instead offers the Fourier spectrum of the centerline and labels it as such; unlike the length-spectrum tone, its pitch is arbitrary, fixed by the parametrization rather than by the knot.
The identification pipeline is a system by W.A.Stanggassinger in collaboration with Claude Code (Anthropic): the sculptor contributes the geometries and decides what to compute; Claude Code writes and runs the code — scanners, invariant layers, Theta integration, and the corpus. The construction method behind the machine families — an atomic module placed under Cn/Sn symmetry with a closure constraint — is named here for attribution and priority; its detailed kinematics are shared with research institutions on request, not published. The symmetry placement shown here is one principle among many: we are developing a broader family of construction machines — building knots from tangles, braids, satellites, woven links and flows, among others. These are exploratory and not yet validated; their constructions are not detailed here and remain work in progress.