Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)
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Eight-crossing state count and trefoil Fox 3-colorings

Example

Assume the Axiom of Countable Choice for the final knot-type distinction. A diagram with c=8 crossings has 28=256 Kauffman bracket smoothing states. For the trefoil diagram below, label its three Fox arcs x1,x2,x3 so the crossing triples are: at C1, over x1 and under x2,x3; at C2, over x2 and under x3,x1; at C3, over x3 and under x1,x2. Thus the crossing equations are 2x1−x2−x3=0, 2x2−x3−x1=0, and 2x3−x1−x2=0. Over F3 the system has nine colorings. The crossingless unknot has one arc and three Fox 3-colorings; because the crossing equation is independent of strand orientations, this count difference distinguishes the underlying knot types.

C1C2C3x1x2x3

The figure is the closure of a two-strand braid with three identical crossings. Each braid strand is oriented downward, and each outside closure arc is oriented upward. At each crossing the upper-left to lower-right strand is continuous and over; the other strand has a gap at the crossing. The labels x1,x2,x3 mark the three arcs between undercrossings, and the crossing incidences used in the statement specify the exact matrix independently of the drawing.

Facts & Assumptions

Given: A nonempty eight-crossing diagram, the stated trefoil incidence data, and p=3.

[F1]

Kauffman bracket state-sum procedure properties: a diagram with c crossings visits exactly 2c smoothing assignments.

[F2]

Quandles and knot-diagram colorings: arcs are cut at undercrossings; a Fox coloring assigns one residue to each arc and obeys the crossing equation encoded by the dihedral operation.

[F3]

Compute dihedral Fox colorings by row reduction: for odd prime p, each crossing contributes 2xo−xi−xj=0 over Fp, and the returned set is parametrized by the free variables of the reduced system.

[F4]

The Fox-coloring algorithm terminates and returns all colorings: if there are f free variables, the procedure returns exactly pf Fox colorings.

[F5]

Dihedral Fox coloring counts are preserved by Reidemeister moves: the Fox p-coloring count is unchanged along every finite sequence of oriented Reidemeister moves and planar isotopies.

[F6]

Reidemeister's theorem for oriented diagrams: under ACω, ambient-isotopic oriented knot diagrams are connected by a finite sequence of those moves.

[A1]

The Axiom of Countable Choice (ACω): ACω is the assumption that every countable family of nonempty sets has a choice function.

Verification

1.1F1givenalgebra

With c=8, the state-count formula in [F1] gives 28=256 visits.

1.2F2F3F4givenalgebra

The stated incidences and [F3] give the equations 2x1−x2−x3=0, 2x2−x3−x1=0, and 2x3−x1−x2=0. In F3, −1=2, so all three rows are (2,2,2). Row reduction leaves one nonzero row and two zero rows, hence one pivot and two free variables. By [F4], the algorithm returns 32=9 trefoil colorings.

1.3F2F4givenalgebra

The crossingless unknot has one arc and no crossing equations, so its matrix has one column and zero rows, rank zero, and one free variable. By [F4], it has 31=3 colorings.

2.1A1F5F6step 1.2step 1.3algebra∎

The crossing equation 2xo=xi+xj is symmetric in the two underarc colors and independent of strand arrows, so reversing a component orientation does not change its Fox coloring set. If the trefoil and unknot were ambient-isotopic as unoriented knots, choose their orientations so the equivalence preserves orientation. By [A1] and [F6] the oriented diagrams would then be joined by a finite Reidemeister sequence, and [F5] would force equal coloring counts. The computed values 9 and 3 contradict this, so the underlying knot types are distinct.

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