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Rasmussen’s s-invariant

Definition

Assume the Axiom of Countable Choice. For an oriented knot $K$ , let $V_{K}$ be its two-dimensional rational Lee homology with the diagram-independent induced decreasing quantum filtration $F^{p} V_{K}$ . For a nonzero class $x \in V_{K}$ , define its filtration level by $ℓ (x) = \max {p \in Z : x \in F^{p} V_{K}} .$ Set $s_{\min} (K) = \min_{x \neq 0} ℓ (x), s_{\max} (K) = \max_{x \neq 0} ℓ (x), s (K) = s_{\max} (K) - 1 = s_{\min} (K) + 1.$ The equality of the two formulas is part of the definition’s well-definedness, supplied by the rank-two Lee filtration theorem.

Definition references

Two-dimensional Lee homology and the two orientation classes

Diagram-independent induced filtration

Rasmussen, Khovanov homology and the slice genus (2004), §3.1

McMullen, 4-Manifolds, §5.6

Facts & Assumptions

Given: ${A C}_{ω}$ and the filtered Lee homology of an oriented knot.

[F1]

Lee homology of a knot is two-dimensional, and the two canonical orientation classes form a basis (Lee homology of a knot has two generators).

[F2]

The induced filtration on Lee homology is independent of the diagram (Lee filtered homology is an oriented link invariant).

[F3]

The two knot filtration levels differ by exactly two, by the marked-circle involution on the two quantum congruence lines (Lee filtration, spectral sequence, and orientation classes).

[A1]

${A C}_{ω}$ is the axiom branch used by the finite diagram-move proof of [F2] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: The hypotheses of the Definition.

1.1F1algebra

The filtration on the finite Lee complex is finite, so its induced filtration on the two-dimensional homology is finite. By [F1] the vector space is nonzero, and therefore every nonzero class has a finite integer level $ℓ (x)$ . The set of levels is a nonempty subset of the finite set of filtration indices; hence its minimum and maximum exist. This uses no choice principle.

2.1A1F1F2F3step 1.1algebra∎

The two extremal filtration levels are the two surviving quantum degrees on the limiting page of the bounded spectral sequence. By [F3], their difference is exactly $2$ , so $s_{\max} - 1 = s_{\min} + 1$ . The diagram-independence conclusion from [F2] uses the stated axiom branch [A1]. Thus the displayed formula defines an invariant of the oriented knot under AC $_{ω}$ .

Depends on

Used by

Rasmussen's invariant for positive knots Corollary
The s-invariant is not twice the knot Floer tau invariant Counterexample
Vanishing tau does not imply smooth sliceness Counterexample
Unknot, trefoils, and the 2-twisted positive Whitehead double Example
Lee cobordism maps give the genus filtration bound Lemma
Lee computations for the Conway knot and its Piccirillo partner Lemma
Rasmussen's invariant changes sign under mirror image Lemma
The Hedden–Ording t=2 positive Whitehead double has tau zero and s two Lemma
Reported refined s-invariant computations Remark
Rasmussen's s-invariant bounds smooth four-ball genus Theorem

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jacob Rasmussen, Khovanov homology and the slice genus, arXiv:math/0402131v1 (standard reference, not scraped)
Curtis T. McMullen, 4-Manifolds, Stanford lecture notes (standard reference, not scraped)