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A fibered knot has rank-one top knot Floer homology
Statement
Assume full AC. Let be a null-homologous knot in a closed connected oriented -manifold , and suppose fibers over with connected fiber a Seifert surface of genus . Then the integral top Alexander group is one copy of the integers:
Statement references
Facts & Assumptions
Actual binding marking and an adapted sutured diagram for a fiber constructs the actual meridionally marked exterior diagram and identifies its fiber cut with the marked product.
Permissible surface niceification and its periodic-domain coefficients gives a finite admissible adapted nice diagram.
Relative sutured diagram moves and their signed equivalences gives the signed comparison for the specified transported orientation class and computes the product by the empty diagram.
Outer fiber grading and the integral cut Euler matrix fixes the absolute Chern anchor: the outer summand has Alexander degree g.
Signed outer summand and cut complex with zero periodic cut identifies the entire outer differential with the cut differential over the integers when the cut periodic group vanishes.
The adapted-admissibility assertion of Signed relative surface cutting and horizontal tensor decomposition, proved by relative winding in step 2.7, makes a good adapted diagram admissible while retaining its decomposing surface and boundary marking. This is the adapted winding operation of Juhasz, Floer homology and surface decompositions, Proposition 4.8; niceification is applied only afterward.
Proof
Given: Full AC, a null-homologous knot K in the stated general closed oriented Y and its connected fiber F of genus g, with the meridional boundary trivialization in the definition of the integral Alexander grading. The Axiom of Choice supplies AC.
Use the actual binding bridge and drilling construction to represent the exterior with two meridional sutures, carrying the actual page longitude and the class of F. The adapted surface cut is literally the result of cutting this mapping torus along its fiber, hence is F times an interval with its product sutures. The fiber has no closed component and its boundary meets both signed boundary regions, so it is a good decomposing surface. First apply the adapted relative winding of F6, away from the protected boundary marking, to obtain an admissible good surface diagram of the same marked exterior and fiber. Now apply the finite permissible niceification, which preserves this established admissibility. Signed relative diagram moves compare the resulting complex to the chosen defining complex, with its specified coherent orientation class transported. No restriction on Y or change of coefficient ring is made.
In this marked adapted diagram the local Euler obstruction is minus 2g plus twice the number of interior polygon coordinates when evaluated on minus F. Therefore the Alexander degree is g minus that number: degree g is precisely the outer generator set, and no higher degree occurs. The product cut has zero periodic-domain group, either directly from its empty product diagram and the relative handle description, or from its vanishing relative integral homology. The signed cut comparison consequently identifies the whole degree-g differential with the cut differential, after harmless changes of generator signs.
The empty diagram of the actual product has one empty-tuple generator and no nonconstant disk, so its integral chain complex is Z with zero differential. The signed relative move equivalences identify its homology with that of the actual cut diagram. Combining with the full top-summand chain identification gives the asserted integral group isomorphism. The exterior orientation character restricts trivially to the zero-periodic cut, so this conclusion holds for the specified defining class without asserting equivalence of arbitrary orientation systems on the uncut exterior.
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Sources
- Yi Ni, Knot Floer homology detects fibred knots, arXiv:math/0607156v4, §1 (standard reference, not scraped)
- Andras Juhasz, Floer homology and surface decompositions, arXiv:math/0609779v3, Theorem 1.5 and §9 (standard reference, not scraped)
- Ciprian Manolescu, An Introduction to Knot Floer Homology, Theorem 2.5 (standard reference, not scraped)