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Tau of the unknot and trefoils from filtered staircases

Example

Assume the Axiom of Countable Choice. Let $U$ be the standard unknot, let $T_{2, 3}$ be the closure of the positive two-strand braid with three crossings, and let $T_{2, 3}^{†}$ be its reverse mirror. In the normalized right-trefoil staircase over $F_{2} [U, U^{- 1}]$ , generators $a, b, c$ have Maslov gradings $0, - 1, - 2$ , Alexander levels $1, 0, - 1$ , and differential $\partial b = U a + c$ , $\partial a = \partial c = 0$ ; multiplication by $U$ lowers both filtrations by one. Choose the polynomial minus lattice $C^{-} = F_{2} [U] ⟨ a, b, c ⟩$ inside this Laurent complex, with each displayed generator at first filtration coordinate zero. Its hat complex is the quotient $C^{-} / U C^{-}$ ; setting $U = 0$ refers to this lattice quotient, not to the Laurent complex in which $U$ is invertible. The full reverse-mirror Laurent staircase is the filtered dual, with generators $a^{\lor}, b^{\lor}, c^{\lor}$ at Alexander levels $- 1, 0, 1$ and differential $\partial a^{\lor} = U b^{\lor}$ , $\partial c^{\lor} = b^{\lor}$ , $\partial b^{\lor} = 0$ . Its chosen polynomial dual lattice has hat quotient with $\partial c^{\lor} = b^{\lor}$ and $\partial a^{\lor} = \partial b^{\lor} = 0$ . The normalized unknot complex has one generator at level zero and zero differential. These models give $τ (U) = 0, τ (T_{2, 3}) = 1, τ (T_{2, 3}^{†}) = - 1.$ Both trefoils have smooth four-ball genus one.

Example references

The normalized filtration threshold defining tau

The one-generator unknot calculation

Duality for the reverse-mirror filtered complex

The smooth four-ball genus bound

The disk-and-band surface and its Euler characteristic

Hom, A survey on Heegaard Floer homology and concordance, §2.2 and §3.1

Ozsvath and Szabo, Knot Floer homology and the four-ball genus, §2.1 and Proposition 3.2

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Chapters 6 and 13

Diagram

The braid schematic is the closure of the positive two-strand braid $σ_{1}^{3}$ . Both strands are oriented downward; the left-to-right strand passes over at each positive crossing, and the exterior arcs join the top and bottom endpoints. The diagram identifies the trefoil convention only; none of the filtered-complex calculations relies on the drawing.

Facts & Assumptions

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

[A1]

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

[F1]

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 $C F K^{\infty} (T_{2, 3})$ staircase with generators $a, b, c$ , $\partial b = U a + c$ , and Maslov gradings $0, - 1, - 2$ . The displayed Alexander levels are $1, 0, - 1$ . The same section identifies $\hat{C F K} (T_{2, 3})$ as the $i = 0$ filtered complex; Example 3.3 in §3.1 computes $τ (T_{2, 3}) = 1$ , 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.

[F2]

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

[F3]

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 $C F K^{\infty} (- K) ≃ C F K^{\infty} (K)^{*}$ , 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.

[F4]

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 $c = 3$ .

[F5]

External input, authorized by the run-specific source boundary. Hom, §3.1, Theorem 3.1(2), printed p. 8, states $∣ τ (K) ∣ \leq g_{4} (K)$ over the survey's stated $F_{2}$ coefficient convention (printed p. 2).

Proof

Given: The complexes and knots in the Statement.

1.1A1F1givenalgebra

By [F1], the standard right-trefoil Laurent complex has the displayed staircase. In the polynomial lattice $F_{2} [U] ⟨ a, b, c ⟩$ , quotienting by $U$ leaves $\partial b = c$ and $\partial a = \partial c = 0$ . The subcomplex at filtration at most $0$ is $⟨ b, c ⟩$ , which is acyclic, while $a$ first appears at level $1$ and represents the total homology class. Thus the normalized threshold is $1$ , as in Hom's Example 3.3.

2.1F1F2F3step 1.1algebra

The dual differential and negated filtration from [F3] give $\partial a^{\lor} = U b^{\lor}$ , $\partial c^{\lor} = b^{\lor}$ , and $\partial b^{\lor} = 0$ for the reverse mirror. In its polynomial dual lattice, setting $U = 0$ leaves $\partial c^{\lor} = b^{\lor}$ and the first nonzero total class occurs at level $- 1$ . The unknot value is [F2].

3.1F4F5step 1.1step 2.1constructalgebra∎

The oriented smoothing of the positive two-strand three-crossing braid has two Seifert circles and three bands. By [F4] it gives a genus-one spanning surface for the positive trefoil; reflecting this surface gives a genus-one spanning surface for its reverse mirror. Thus each four-ball genus is at most one. The thresholds in steps 1.1–2.1 and Hom's genus bound [F5] give the lower bound one for each; hence both smooth four-ball genera equal one.

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Sources

Jennifer Hom, A survey on Heegaard Floer homology and concordance (standard reference, not scraped)
Peter S. Ozsvath and Zoltan Szabo, Knot Floer homology and the four-ball genus (standard reference, not scraped)
Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links (standard reference, not scraped)