Knot Floer genus and fibering detection — Examples and Counterexamples

Given: AC, the Poincaré sphere $P$, the connected sum $Y=P\mathrm{\#}P$, a connected-sum sphere $S$, a standard round unknot $U$ in a $3$-ball $B_0$ contained in the first summand and disjoint from $S$, and a tubular neighborhood $N(U)\subset B_0$.

The Poincaré sphere is a nontrivial integral homology sphere: $H_1(P;\mathbb{Z})=H_2(P;\mathbb{Z})=0$ and ${\pi }_1(P)\mathrm{\ne }1$.

Mayer–Vietoris sequence in singular homology: for an open cover, the sequence $H_1(U\cap V)\to H_1(U)\oplus H_1(V)\to H_1(X)\to H_0(U\cap V)\to H_0(U)\oplus H_0(V)$ is exact.

Alexander duality for compact locally contractible subsets of a sphere, Homology of spheres, and The universal coefficient theorem for cohomology over a PID: $H_1(S^2;\mathbb{Z})=0$; for an unknot $U\cong S^1$ in $S^3$, ${\tilde{H}}_0(S^3\setminus U;\mathbb{Z})=0$ and ${\tilde{H}}_1(S^3\setminus U;\mathbb{Z})\cong {\tilde{H}}^1(U;\mathbb{Z})\cong \mathbb{Z}$.

Poincaré duality for oriented topological manifolds and The universal coefficient theorem for cohomology over a PID: under AC, a closed connected oriented $3$-manifold with $H_1=0$ has $H_0=H_3=\mathbb{Z}$ and $H_2\cong H^1=0$.

Every spherical gluing below is made with a retained product collar $S^2\times (-2\epsilon ,2\epsilon )$, chosen disjoint from $U$. For closed pieces $A_{-}$ and $A_{+}$ on the negative and positive sides, replace them by the relative-open sets $O_{-}=A_{-}\cup (S^2\times (-2\epsilon ,\epsilon ))$ and $O_{+}=A_{+}\cup (S^2\times (-\epsilon ,2\epsilon ))$ in their union. On the added collar of $O_{-}$, send its parameter $t\ge 0$ to $(1-s)t$; fix $A_{-}$. The analogous formula on the negative added collar retracts $O_{+}$ to $A_{+}$. Their intersection is $S^2\times (-\epsilon ,\epsilon )$ and retracts to $S^2$ by $t\mapsto (1-s)t$. These homotopies match the identity on the joining sphere and remain valid after deleting $U$, since the collars miss it. Thus [F2] is applied only to these actual open covers, with the same homology groups as the named closed pieces. The collars are the product collars retained in the chosen smooth ball and connected-sum constructions; no collar for an arbitrary wild sphere is asserted.

The Axiom of Choice: AC is the assumption of the local Poincaré-duality suppliers.

Given: Full AC, the right-handed trefoil and standard alternating knot $5_2$ with the original integral normalization and classical choice branch declared in the dependencies.

External input. Ozsváth–Szabó, Heegaard Floer homology and alternating knots, Geometry & Topology 7 (2003), Theorem 1.3, printed p.230 (PDF p.6 of arXiv:math/0209149v3): if $K$ is alternating and its normalized symmetric Alexander polynomial is $a_0+{\sum }_{s>0}{a}_s(t^s+t^{-s})$, then $\widehat{HFK}(S^3,K,s;\mathbb{Z})\cong {\mathbb{Z}}^{\mathrm{\mid }{a}_s\mathrm{\mid }},$ supported in Maslov degree $s+\sigma (K)\mathrm{/}2$. This is the INTEGRAL theorem, including freeness; no integral conclusion is inferred from the retained mod-two grid comparison. Its signature convention has left trefoil signature $+2$.

External finite table input. Knot Atlas, pages 3_1 and 5_2, section “Polynomial invariants / Alexander polynomial”, records respectively $t-1+t^{-1}$ and $2t-3+2t^{-1}$. These two finite entries and their standard knot diagrams identify the table knots. The same polynomials result from the standard genus-one disk/band Seifert matrices $V_3=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),V_5=\left(\begin{array}{cc}1& 1\\ 0& 2\end{array}\right)$ (up to simultaneous handedness/sign conventions): $\det (V_3-t{V}_3^T)=1-t+t^2$ and $\det (V_5-t{V}_5^T)=2-3t+2t^2$. Multiplying by $t^{-1}$ gives the stated symmetric polynomials. The TABLE/diagram identification is external input; the two determinant calculations are finite local checks, not an assertion that an abstract matrix by itself identifies a knot.

External input at the exact $S^3$, integral scope. Manolescu, An Introduction to Knot Floer Homology, printed/PDF p.5, Theorem 2.3, gives $g(K)=\max \{s:\widehat{HFK}(K,s;\mathbb{Z})\mathrm{\ne }0\}$ for every knot in $S^3$. Theorem 2.5 on that same page gives that a knot in $S^3$ is fibered if and only if its integral top genus group is $\mathbb{Z}$. These exact specializations of the established Ozsváth–Szabó/Ghiggini/Ni/Juhász results are imported. This does not assert completion of the held local general-$Y$ signed comparison or product-criterion proofs. The source explicitly states these as integral groups.

Actual reading limits: the complete Theorem 1.3 statement and its preceding conventions on printed pp.229–230 of the primary alternating paper; the two named Alexander-polynomial table entries (not all table invariants); the full Theorems 2.3 and 2.5 statements and surrounding geometric-applications text on Manolescu p.5. No reading or local reconstruction of the full alternating theorem proof, Ni proof, or entire survey is claimed. These are explicitly authorized external imports under the run-specific source boundary. The original full-choice branch is retained; no integral full-choice theorem is silently cast as a CC theorem.