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Rank-one top knot Floer homology detects fibering

Statement

Assume the Axiom of Choice. Let $K$ be a null-homologous knot in a closed connected oriented $3$ -manifold $Y$ , and suppose that the knot complement $Y ∖ K$ is irreducible. Let $F$ be a genus- $g$ Seifert surface for $K$ . If the integral top Alexander group satisfies

$\hat{H F K} (Y, K, [F], g; Z) ≅ Z,$

then $Y ∖ K$ fibers over $S^{1}$ with fiber $F$ .

Statement references

The integral top Alexander group

The sutured complement cut along F and its product structure

Ni, Knot Floer homology detects fibred knots, Theorem 1.1 and §§3–7

Juhasz, Floer homology and surface decompositions, Theorem 9.11

Facts & Assumptions

Given: Full AC, the knot, exterior, actual surface $F$ and integral grading- $g$ group in the Statement.

[F1]

Under full AC and exactly these geometric and integral-group hypotheses, Ni's sutured Floer and Gabai product criterion proves that the compact exterior cut along the actual $F$ is a product sutured manifold. Its proof explicitly imports the general balanced sutured vanishing and nonproduct rank results and records Ni's corrected characteristic-pair formulation; the resolved local signed comparison and hierarchy dependencies are preserved.

Proof

technique · direct

1.1F1

Apply F1 to the actual class $[F]$ , the actual genus $g$ and the given integral group. The cut exterior is a product $F \times [0, 1]$ as a sutured manifold. In particular the two horizontal faces correspond to the two copies of the original $F$ , rather than to a surface selected only by a genus or homology calculation.

2.1step 1.1

To restore the exterior, identify those two faces using their actual gluing map. In product coordinates this is $(x, 1) \sim (h (x), 0)$ for a self-homeomorphism $h : F \to F$ . Thus the exterior is the mapping torus of $h$ . Its projection to $[0, 1] / (0 \sim 1) = S^{1}$ is a surface bundle, and the seam is precisely the original embedded $F$ .

3.1step 2.1∎

The vertical boundary is the mapping torus of $\partial F$ . Its boundary collar extends this bundle over the deleted-knot end, giving the fibration of $Y ∖ K$ with fiber the interior of $F$ , or equivalently with compact fiber $F$ in the compact exterior. This proves the stated fiberedness with the prescribed surface, for arbitrary $Y$ and also for genus zero.

Depends on

Used by

Trefoil and 5₂ knot Floer genus and fibering cases Example

Dependency tree · two levels

18 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

Yi Ni, Knot Floer homology detects fibred knots, arXiv:math/0607156v4, Theorem 1.1 (standard reference, not scraped)
Andras Juhasz, Floer homology and surface decompositions, arXiv:math/0609779v3, Theorem 9.11 and proof (standard reference, not scraped)