ExampleConstruction: Literature-sourcedVerification: Not suppliedPipeline-generatedverified 2026-10-22 (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.

Worked grid-complex cases for the unknot

Example

Assume countable choice. Let $U$ be the oriented unknot and let $G_{2}, G_{3}$ be the specified $2$ -by- $2$ and $3$ -by- $3$ grids. Their tilde homology ranks are $2$ and $4$ . For each of these grids, the grid-to-knot-Floer identification gives $\tilde{G H} (G_{n}) ≅ \hat{H F K} (U) \otimes V^{\otimes (n - 1)}$ , where $n = 2$ or $3$ , $V$ has one generator in bidegree $(0, 0)$ and one in bidegree $(- 1, - 1)$ , and $\hat{H F K} (U)$ has rank one by the standard genus-zero diagram and pointed-diagram invariance. Thus the respective ranks are exactly the dimensions supplied by the stable factors $V$ and $V^{\otimes 2}$ .

Example references

The 2-by-2 finite rectangle calculation

The 3-by-3 finite rectangle calculation

The genus-zero one-pointed unknot calculation

The grid-to-HFK identification and stable factor

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Exercise 4.4.3 and Proposition 4.6.15

Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §§3–4

Depends on

Used by

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Sources

Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links, Exercise 4.4.3 and Proposition 4.6.15 (standard reference, not scraped)
Ciprian Manolescu, Peter Ozsvath, and Sucharit Sarkar, A combinatorial description of knot Floer homology, §§3–4 (standard reference, not scraped)