CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

HHH is an oriented-link invariant up to an overall trigrading shift

Statement

Assume the Axiom of Choice. Let $D_{1}, D_{2}$ be braid diagrams with nonempty closures that are ambient-isotopic oriented links in $S^{3}$ (Oriented links in the three-sphere and ambient isotopy, The closure of a geometric braid). Then there is a trigrading shift $(h_{0}, p_{0}, c_{0}) \in Z^{3}$ , depending only on the two diagrams and the chosen sequence of Markov moves (Markov conjugation and stabilization moves), such that, with $H H H^{c, h, p} = 0$ for $h < 0$ when translating the grading, $H H H^{c, h, p} (D_{1}) ≅ H H H^{c + c_{0}, h + h_{0}, p + p_{0}} (D_{2}) for all h, p, c;$ that is, the trigraded HHH of a braid diagram is an invariant of the oriented link $\hat{D}$ up to an overall trigrading shift. If the Khovanov-Rozansky shift of the chosen Markov sequence is $(j_{0}, k_{0}, l_{0})$ , the corresponding HHH shift is $c_{0} = j_{0}, h_{0} = - k_{0}, p_{0} = l_{0} - k_{0} .$ In particular the type IA stabilization, which contributes ${1, 1} [1]$ on the Khovanov-Rozansky side, contributes $(h_{0}, p_{0}, c_{0}) = (1, 0, 1)$ to the HHH trigrading, while the type IB stabilization contributes no shift.

Caveats: the Axiom of Choice enters through Markov's closed-braid equivalence theorem and the diagonal Koszul comparison used by the HHH theorem, and the homogeneous basis choices in the inherited reduced-invariance argument; the shift is not canonical without fixed conventions, and the HHH shift is only determined by the same Markov sequence as the Khovanov-Rozansky one, so no absolute normalization is silently added; the statement is about the reduced theory.

Facts & Assumptions

Given: braid diagrams $D_{1}, D_{2}$ whose closures are ambient-isotopic oriented links, and AC.

[F1]

Markov's theorem: two braid closures are ambient-isotopic oriented links if and only if the braids are related by a finite sequence of Markov moves (conjugation, braid-group transformations, stabilization/destabilization), under AC (Markov's theorem for braid closures).

[F2]

If the closures of $D_{1}$ and $D_{2}$ are ambient-isotopic, then there is a trigrading shift $(j_{0}, k_{0}, l_{0})$ with ${\overline{H}}_{k, l}^{j} (D_{1}) ≅ {\overline{H}}_{k + k_{0}, l + l_{0}}^{j + j_{0}} (D_{2})$ for all $j, k, l$ , by the reduced-invariance proof of The reduced Khovanov-Rozansky homology; the shift is the product of the shifts of the Markov sequence, the type IA stabilization contributing ${1, 1} [1]$ and the type IB stabilization none (Khovanov-Rozansky braid homology is an oriented link invariant up to shift).

[F3]

The comparison theorem gives a trigraded isomorphism between the termwise Hochschild theory of a braid word and the reduced Khovanov-Rozansky homology of its closure, with $k = - h$ , $l = p - h$ , $j = c$ after the global correction $(k, l) \mapsto (k + 1, l - 1)$ (HHH is isomorphic to reduced Khovanov-Rozansky homology, The Koszul-Hochschild comparison respects crossing differentials and trigradings).

[F4]

$D_{1}, D_{2}$ are braid diagrams of oriented links with the same closure class, so their closures are ambient-isotopic oriented links (Oriented links in the three-sphere and ambient isotopy, The closure of a geometric braid).

[F5]

AC is the choice-function principle (The Axiom of Choice), used through [F1], [F2], and [F3].

Proof

technique · direct

1.1F1F2F4givenalgebra

Markov sequence and the Khovanov-Rozansky shift. Since the closures of $D_{1}$ and $D_{2}$ are ambient-isotopic oriented links by [F4], [F1] provides a finite sequence of Markov moves connecting the two braid words; by [F2] and The reduced Khovanov-Rozansky homology the reduced trigraded groups satisfy ${\overline{H}}_{k, l}^{j} (D_{1}) ≅ {\overline{H}}_{k + k_{0}, l + l_{0}}^{j + j_{0}} (D_{2})$ for the shift accumulated along that sequence. This is an isomorphism of the reduced graded vector spaces, which is the input needed for the comparison theorem.

2.1F3step 1.1algebra

Transport through the comparison. By [F3] each side is identified with $H H H$ under the dictionary after the correction: a Khovanov-Rozansky class of tridegree $(j, k, l)$ corresponds to the HHH class $(c, h, p) = (j, - k, l - k)$ when expressed in corrected gradings. Conjugating the isomorphism of step 1.1 by the identifications of [F3] therefore gives $H H H^{c, h, p} (D_{1}) ≅ H H H^{c + c_{0}, h + h_{0}, p + p_{0}} (D_{2})$ with $c_{0} = j_{0}$ , $h_{0} = - k_{0}$ and $p_{0} = l_{0} - k_{0}$ : indeed a shift by $(j_{0}, k_{0}, l_{0})$ of $(j, k, l)$ changes $(c, h, p) = (j, - k, l - k)$ to $(j + j_{0}, - k - k_{0}, l - k + l_{0} - k_{0}) = (c + c_{0}, h + h_{0}, p + p_{0})$ with the displayed values.

3.1F1F2F5step 2.1algebra∎

Stabilization shifts and conclusion. The type IA stabilization contributes ${1, 1} [1]$ by [F2], from the straight diagram $D_{1}$ to the curl $D_{2}$ in the source's convention. Since $(M {r})_{k} = M_{k - r}$ and $H^{j} (C [1]) = H^{j + 1} (C)$ , the object equality $C (D_{1}) = C (D_{2}) {1, 1} [1]$ gives group indices $(j_{0}, k_{0}, l_{0}) = (1, - 1, - 1)$ . Hence its HHH group shift in the displayed direction is $(c_{0}, h_{0}, p_{0}) = (1, 1, 0)$ , and the type IB stabilization contributes $(0, 0, 0)$ . The composition of the shifts along the Markov sequence is associative because each shift is the multiplication of the grading by a fixed translation, so the total HHH shift depends only on the two diagrams and the chosen sequence, exactly as the Khovanov-Rozansky shift does. The Axiom of Choice is inherited in step 1.1 through Markov's theorem [F1] and in step 2.1 through the diagonal comparison in [F3] and the homogeneous basis choices for reduced invariance in [F2]. The reverse stabilization reverses all three group-index shifts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, arXiv:math/0510265v3 (19 printed pages); Theorem 1 (standard reference, not scraped)
Mikhail Khovanov and Lev Rozansky, Matrix factorizations and link homology II, arXiv:math/0505056v2 (37 printed pages); Theorem 1 and Theorem 2 (standard reference, not scraped)
Eugene Gorsky, Oscar Kivinen and Jose Simental, Algebra and geometry of link homology: Lecture notes from the IHES 2021 Summer School, Bull. London Math. Soc. 55 (2023) 537-591; section 3 (standard reference, not scraped)