The Knot Tabulator — knot determinant

Wiggle it all you like. Encode a diagram as a signed Gauss code, compute its knot determinant |Δ(−1)|, then apply Reidemeister moves at random: under every move the number can't budge. The trefoil reads 3, the figure-8 5, the unknot 1 — proof a tangled loop is truly knotted. Ariadne draws plaits but never proves the knot knotted; this is her rigorous twin.

Specimen

Unknot Trefoil Figure-8 Hopf link

The invariant — what cannot move

determinant |Δ(−1)| —

—

p-colourings (nontrivial)—

3-colourable?—

The big number is whatever the core returns — never hardcoded. It is recomputed from the live Gauss code every move; during an animation its DOM is not touched.

The picture — what may move

crossings drawn—

Gauss tokens—

Wiggle

Wiggle it ⟳ ▶ auto ↺ reset

One press = one random valid Reidemeister move (the core chooses & reports it; the picture can't lie). Watch the diagram morph while the determinant holds byte-still.

View

Diagram

3-colouring

In 3-colouring mode each arc is stroked by a legal Fox colour. The trefoil shows all three; the unknot visibly can't be coloured non-trivially.

The teeth — a fake invariant bites

Apply an R1 twist ↻

The same R1 move, but watch the raw crossing count (a FAKE invariant) tick up n→n+1 while the real |Δ| holds steady. The negative control bites — so the stability above is not vacuous.

Legend

the strand (the knot itself)

a 2nd component (a link = two hues — Hopf)

the 3 Fox colours (colour-blind-safe)

What this bench proves — & doesn't

PROVEN here: stability (the determinant is byte-identical on a random Reidemeister walk over THESE diagrams — not a proof for all knots), discrimination (det 3≠1; trefoil 3-colourable, the unknot not), and the teeth (crossing count changes under R-I).

Stated, not re-derived: invariance under ALL Reidemeister moves for ALL diagrams is a theorem (Alexander/Goeritz); the bench CHECKS it on a random walk over 4 diagrams.

The one-way caveat: det ≠ 1 ⟹ knotted (sound — the unknot has det 1). The converse is FALSE — some knots also read 1 — so this proves knottedness, never unknottedness. The determinant is also blind to mirror image (left & right trefoil both read 3): it proves a loop knotted, it does not classify it.

Scope: one bench, one invariant, four hard-coded diagrams + a verified move-applier — deliberately not a freehand editor or a topology wing. Jones polynomials, the knot group, draw-your-own: each a fresh seed, not an in-place expansion.