The tau invariant and four-ball genus — Examples and Counterexamples

Given: The standard unknot, the positive right-handed trefoil, its reverse mirror, and Countable Choice.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is the assumption retained in the Statement.

External input, authorized by the run-specific source boundary. Hom, A survey on Heegaard Floer homology and concordance, §2.2, Figure 3(a),(c), gives the normalized $CF{K}^{\mathrm{\infty }}(T_{2,3})$ staircase with generators $a,b,c$, $\mathrm{\partial }b=Ua+c$, and Maslov gradings $0,-1,-2$. The displayed Alexander levels are $1,0,-1$. The same section identifies $\widehat{CFK}(T_{2,3})$ as the $i=0$ filtered complex; Example 3.3 in §3.1 computes $\tau (T_{2,3})=1$, and Theorem 3.1(1) states that this is the standard knot invariant. I read the figure caption, threshold definition, and examples; I do not claim to reconstruct the Heegaard-diagram computation.

The tau level of the standard unknot is zero computes $\tau (U)=0$ in the normalized genus-zero one-pointed model.

External input, authorized by the run-specific source boundary. Hom, The knot Floer complex and the smooth concordance group, §2, PDF/printed p. 5, defines $-K$ as the reverse mirror and states $CF{K}^{\mathrm{\infty }}(-K)\simeq CF{K}^{\mathrm{\infty }}(K{)}^{\ast }$, with arrows and both filtrations reversed. Its dual-basis description gives the displayed reverse-mirror staircase from [F1]. The finite filtered hat threshold sign is also proved for the supplied diagrams by The normalized filtered complex of a reverse mirror is dual. I read the duality statement and its convention; I do not claim to reconstruct its cited Heegaard-move proof.

Euler characteristic and genus of a Seifert disk-and-band surface computes the genus of the Seifert disk-and-band surface from its circle and crossing counts; for the displayed positive three-crossing braid the counts are $s=2$ and $c=3$.

External input, authorized by the run-specific source boundary. Hom, §3.1, Theorem 3.1(2), printed p. 8, states $\mathrm{\mid }\tau (K)\mathrm{\mid }\le g_4(K)$ over the survey's stated ${\mathbb{F}}_2$ coefficient convention (printed p. 2).

Given: The Axiom of Countable Choice, the universal implication, and the specified positive-clasped, two-full-twist Whitehead double in Hedden and Ording's convention.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) is the axiom assumption retained in the Statement.

External input, authorized by the run-specific source boundary. Hedden and Ording, The Ozsváth–Szabó and Rasmussen concordance invariants are not equal, Theorem 1.1, printed p. 2, states $\tau (D_{+}(T_{2,3},2))=0$ and $s(D_{+}(T_{2,3},2))=2$ for the two-twisted positive Whitehead double of the right-handed trefoil. Their Figure 1 specifies the twist box as full right-handed twists. The theorem states both values. Section 2.1, printed p. 4, reports a computer-computed Khovanov Poincaré polynomial: its homological-degree-zero support forces $s\in \{2,4\}$, and the genus-one upper bound together with Rasmussen's inequality forces $s=2$. The acknowledgments on printed p. 3 credit Rasmussen with the $s$ computations. This is a published derivation using computer output, not a reproduced Lee chain-complex computation; the numerical theorem is imported here.

External input, authorized by the run-specific source boundary. Rasmussen, Khovanov homology and the slice genus, Theorem 1 in §1, PDF p. 1, states $\mathrm{\mid }s(K)\mathrm{\mid }\le 2g_4(K)$ for every knot, where $g_4$ is the minimum genus of a smooth properly embedded surface in $B^4$. The source proof in §4.4 derives this from the filtered Lee cobordism map; I read that proof. This is the same smooth genus convention as Smooth and topological slice genus.

The Alexander filtration and tau level uses the same normalized threshold convention for the standard Ozsváth–Szabó invariant in [F1]. The target total homology is one-dimensional, so nonzero image agrees with Hom's surjective-inclusion convention.

The $s$ in [F1] is Rasmussen's standard integer obtained from rational Lee homology with the convention $s=s_{\max }-1$ stated in the Statement. The external Theorem 1 in [F2] uses this normalization; no local Lee-map proof is used here.

Smooth and topological slice genus defines $g_4^{\mathrm{s}\mathrm{m}}$ by minimizing the genera of smooth connected oriented properly embedded surfaces in $B^4$ with boundary the knot.

External input, authorized by the run-specific source boundary. Hom, A survey on Heegaard Floer homology and concordance, §3.1, Theorem 3.1(1), printed p. 8, identifies the standard $\tau$ defined by the same normalized threshold as a knot-concordance homomorphism. The survey's coefficient convention is stated on printed p. 2. I read the threshold definition and theorem statement; I do not claim to reconstruct the cited proof.