LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The standard genus-zero hat diagram of the unknot has rank one

Statement

Let $U$ be the standard oriented unknot and take its genus-zero one-pointed Heegaard diagram: $Σ = S^{2}$ , each handlebody is a three-ball, there are no $α$ - or $β$ -curves, and one $w$ -point and one $z$ -point are joined by trivial arcs in the two balls. The associated-graded hat chain complex has one generator, zero differential, and homology of rank one over $F_{2}$ . This is a computation for the stated diagram; it makes no invariance claim for other diagrams.

Statement references

The pointed Heegaard diagram and associated-graded hat complex

The oriented unknot in the standard three-sphere

Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §2

Manolescu, An Introduction to Knot Floer Homology, §3.5

Facts & Assumptions

Given: The genus-zero, one-pointed diagram specified in the statement.

[F1]

Admissible multipointed Heegaard data for knot Floer complexes defines ${Sym}^{m} (Σ)$ , the product tori, and their intersection points as generators; its hat specialization sets the basepoint variable to zero.

[F2]

Admissible multipointed Heegaard data for knot Floer complexes defines the differential by counts of classes of Maslov index $1$ .

Proof

1.1F1givenconstruct

Here $h = 0$ and $k = 1$ , so $m = h + k - 1 = 0$ . The empty products of $α$ - and $β$ -curves both map to the unique point of ${Sym}^{0} (S^{2})$ , which is a singleton. By [F1], this intersection is the entire generator set, so the chain group has dimension one.

2.1F1F2step 1.1algebra∎

Every map into a singleton is constant, so there is only the constant class in $π_{2} (x, x)$ . Its pullback tangent bundle is zero-dimensional; hence the linearized Cauchy–Riemann operator has zero kernel and cokernel and Maslov index $0$ . Thus there is no class of Maslov index $1$ to contribute under [F2]. The differential is zero, and the homology equals the one-dimensional chain group.

Depends on

Used by

Raw tilde grid homology depends on the unknot grid Counterexample
Tau of the unknot and trefoils from filtered staircases Example
Worked grid-complex cases for the unknot Example
The tau level of the standard unknot is zero Lemma
Invariance of the multipointed Heegaard Floer complex Theorem

Dependency tree · two levels

7 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

Ciprian Manolescu, Peter Ozsvath, and Sucharit Sarkar, A combinatorial description of knot Floer homology, §2 (standard reference, not scraped)
Ciprian Manolescu, An Introduction to Knot Floer Homology, §3.5 (standard reference, not scraped)