Knot Floer homology detects the Seifert genus

Choose a Seifert surface F realizing the least genus g=g(K), and let E be the compact exterior. The exterior is irreducible: a sphere in its interior bounds two balls in S³ by F1; the connected knot neighborhood lies entirely on one side, so the other ball is disjoint from it and lies in E. A small oriented meridian loop meets F once, so F is nonseparating: a separating two-sided surface would have algebraic intersection zero with every closed loop. Cut E along the connected F to obtain connected M with the product annular suture and the two F faces. Their Euler characteristics agree and the boundary meets that suture, so M is balanced. A sphere in M is disjoint from F in E. Its E-ball is also disjoint from F: since F is connected, disjoint from the sphere, and has its boundary on ∂E outside that interior ball, it lies entirely on the other side. Thus that ball remains a ball in M. This proves irreducibility of the actual cut without a general-Y irreducibility assertion.

The two faces are incompressible. A compression of F along a nonseparating essential circle produces a connected Seifert surface of genus g−1. A separating compression produces a boundary-bearing Seifert component and a closed component. The latter has positive genus, since otherwise the original compression circle bounded a disk on F; discarding it again gives a Seifert surface of genus strictly less than g. Both contradict minimality. A face compression in M would be such a compression in E. For g=0 the disk has no essential compression circle.

To prove the actual sutured norm-minimality, take any compact oriented properly embedded competitor T representing the positive face class in H₂(M,γ;Z). Its total oriented boundary in H₁(γ;Z) is one copy of the longitude. Put its boundary circles in the interior of the annular suture. Cap an innermost contractible boundary circle by its disk in the annulus, pushed into the boundary collar; this operation increases Euler characteristic and cannot increase the Thurston complexity. Continue until every boundary circle is essential. Whenever both orientations remain, choose an adjacent opposite pair. The annulus between them contains no other boundary circle. Glue it to T in a thin boundary collar and round the corners. This gives an embedded oriented surface and removes the pair. Euler characteristic is unchanged; if two components merge, the inequality max(0,−(χ₁+χ₂))≤max(0,−χ₁)+max(0,−χ₂) shows that total complexity cannot increase. For two circles on one component its Euler characteristic, hence complexity, is unchanged. The finite sequence ends with exactly one essential boundary circle, since its total winding is one. Delete all closed components, which also cannot increase complexity. The remaining connected surface embeds in E and extends across the knot collar to an actual Seifert surface for K of the same genus. Its genus is at least g by minimality. For g≥1 its complexity is therefore at least 2g−1=x(F); for g=0 it is at least zero=x(F). It follows that x(T)≥x(F). The same argument applies to the negative face. Thus M is the actual taut balanced sutured manifold, by irreducibility and norm-minimality. The boundary cancellations were used only to derive a lower bound and need not preserve the competitor's relative class after it is mapped to E.

Work under the full-Choice analytic framework F6 on the actual fixed finite hierarchy and marked adapted diagram. Its admitted sutured operator problems retain the boundary/divisor barriers, periodic-domain and area conditions and the specified coherent orientation class. With the regular data and compactification/gluing construction supplied at that scope, apply F3 to the marked Seifert surface F. Its meridional frame, rather than an unspecified closure framing, gives an admissible adapted representative. By F4 every generator has Alexander degree g−n_P(x), with n_P(x) a nonnegative integer. Hence its whole integral chain complex has no generator, and thus no homology, in a grading greater than g.

The entire degree-g summand is the outer summand, and F3 identifies its signed differential with the actual cut complex of M. Since M is taut, F2 gives that this cut homology contains a split copy of Z for the orientation class induced by the given knot complex. Consequently the integral knot Floer group at degree g is nonzero. This also covers g=0: the taut cut argument and the same signed chain map apply to the minimum disk, without a genus-one connected-sum or field-coefficient substitution.

Degree g is nonzero and every larger degree is zero, so the maximum of the stated nonvanishing degree set is exactly g(K). The proof uses only the literal S³ ball-side and minimum-Seifert-genus arguments above; it makes no claim of arbitrary general-Y Thurston norm detection.