ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Why the Khovanov-Rozansky II invariance proof stays in the braid-diagram calculus

Example

Assume AC (The Axiom of Choice) for the sourced Markov comparison of ambient-isotopic closures. Tabulate the moves actually used in the invariance proof of Khovanov-Rozansky braid homology is an oriented link invariant up to shift:

(i) the braid-like Reidemeister IIa move and far commutations; (ii) the braid-like Reidemeister III move with coherent orientations; (iii) conjugation of braid words; (iv) the two oriented stabilizations/destabilizations via the type IA and IB kink computations; (v) Markov's equivalence theorem Markov's theorem for braid closures.

The oriented Reidemeister IIb move is not in the list, and Khovanov-Rozansky II, printed p. 9 states explicitly that the authors did not prove IIb invariance and restricted to braid diagrams to avoid it; for closed braids the IIb move is never needed, because Markov's theorem accounts for all isotopies of the closures. This example illustrates proof scope only; it makes no claim that the complex fails to be invariant under IIb, and it does not claim IIb invariance. Caveat: the type IA and IB pictures must be the source's oriented pictures, since the stabilizations carry different shifts.

Facts & Assumptions

Given: AC, the invariance theorem's proof, its list of Markov moves, and the source's statement of the IIb obstruction.

[F1]

The braid-homology invariance theorem reduces the proof to the three Markov moves and obtains the invariance from the IIa move, the coherent-orientation III move, conjugation, the two oriented kink computations and Markov's theorem; the Axiom of Choice enters only through Markov's theorem (Khovanov-Rozansky braid homology is an oriented link invariant up to shift).

[F2]

Markov's theorem for braid closures: two closed braids are ambient-isotopic oriented links if and only if the braids are related by conjugation and stabilization/destabilization moves; in particular no Reidemeister move outside the braid-diagram calculus is needed for the comparison of closures (Markov's theorem for braid closures).

[F3]

AC is the choice-function principle, assumed for the Markov theorem used here (The Axiom of Choice).

Verification

technique · comparison of the proof's move list with the source's stated scope; no new mathematics

1.1F1F2

The moves used, with their roles. In the proof of [F1] far commutations $σ_{i} σ_{j} \leftrightarrow σ_{j} σ_{i}$ are the signed tensor flips of disjoint crossing factors, the braid-like IIa move covers inverse cancellations $σ_{i} σ_{i}^{- 1}$ , $σ_{i}^{- 1} σ_{i}$ ; the coherent-orientation III move covers the braid relation; conjugation covers the change of cyclic order; the kink computations cover the two oriented stabilizations/destabilizations with their shifts ${1, 1} [1]$ and none; and marking changes are absorbed by the marking-independence theorem. These are the source's Markov moves (a)-(c), with marking independence supplying the auxiliary marking changes. The separate six defining properties of $F$ include a skein relation and an unknot normalization, so they are not a move list.

1.2F1F2F3

Why IIb is absent and why this is not a gap. The source restricts the construction to braid diagrams because the oriented IIb move lies outside the braid-diagram calculus and its invariance is not proved there. By [F2], an ambient isotopy between two braid closures can be replaced by a finite Markov sequence, so IIb is never invoked in the invariance argument; conversely the example claims no invariance under IIb and no failure of it, only that the proof's scope is the braid-diagram calculus. The only choice-theoretic input is Markov's theorem inside [F1], as recorded there.

2.1F1F2step 1.1∎

Conclusion. The tabulation of step 1.1 is complete: every move used in the invariance proof is one of far commutation, braid-like IIa cancellation, coherent-orientation III, conjugation, the two oriented stabilizations, or a marking change composed along a Markov sequence, and the IIb move is neither used nor claimed. This is a scope illustration, not a counterexample or a failure statement.

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