Analytic inputs for Ni's sutured knot Floer decompositions

Construct the actual marked closure/adapted surface diagram from the relative Morse handles, carrying the page longitude and meridional boundary frame. Positive boundary fingers, the uniform relative windings, and permissible niceification give the admitted diagram. The Chern obstruction formula evaluates on minus F as minus 2g plus twice the number of interior polygon coordinates. Consequently its degree-g generators are exactly its outer generators. Each index-one disk is an exhausted embedded bigon or rectangle and lifts uniquely in the cut/double surface. The literal Fredholm conjugacies and the pullback of the specified periodic-domain orientation character align every coefficient after generator signs. Thus this generator correspondence identifies the full disk-count differentials over the integers. Signed relative moves transport the identification to the defining complexes, with their markings and coherent orientation classes.

For a horizontal decomposition, the cut components are balanced. In its adapted diagram there is no reverse cross-family intersection from side two to side one; the equal row and column counts force every perfect-matching generator to avoid the polygon. Thus the whole complex is the outer complex. The cut surface is a disjoint union, so each index-one disk has exactly one nonconstant factor. Ordering determinant factors gives the Koszul differential ∂(x1⊗x2)=∂x1⊗x2+(−1)^|x1|x1⊗∂x2. This is the horizontal chain map and the complete integral chain formula. Taking homology over any field gives Ni's horizontal formula. Taking homology over the integers instead gives the Künneth exact sequence including Tor; no unqualified integral homology tensor formula is asserted.

For an essential separating product annulus with connected R+ and R− and two balanced cut pieces, form the two rounded horizontal surfaces from opposite boundary faces and the annulus. Their actual cobordism regions give the required relative class and their Euler sums give the horizontal equality. This reduces the operation to attachment of a product region. A rooted cut-arc forest removes a product region meeting the sutures by literal product-disk identifications. For an interior essential product region, the explicit once-punctured-torus block forces its two fixed coordinates and isolates the only possible extra genus-one two-corner domain, whose Maslov index is minus one; it cannot contribute. Both comparison diagrams therefore have precisely the same supported disks and conjugate oriented operators in the unchanged complement. The resulting integral chain identification, followed by the remaining product-disk cuts, gives the full separating-annulus graded tensor operation. It gives Ni's field formula and the integral Künneth/Tor formulation as above. All constructions keep the stated basepoint zero regions, actual periodic lattices and essential-annulus hypotheses. They apply when the specified characteristic product region supplies those actual surfaces; they do not infer its geometry from generic disk existence.