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Lee homology of a knot has two generators

Statement

Assume the Axiom of Countable Choice. Let $K$ be a nonempty oriented knot in the standard oriented three-sphere. Its rational Lee homology is a two-dimensional vector space. For every finite regular diagram of $K$ , the two homology classes obtained from the two orientations by the oriented-resolution parity-of-nesting rule form a basis, and the induced quantum filtration is independent of the diagram.

Statement references

Canonical Lee orientation classes and their basis theorem

Diagram-independent filtered Lee homology

Rasmussen, Khovanov homology and the slice genus (2004), §§2.4 and 3.1

McMullen, 4-Manifolds, §§5.5–5.6

Facts & Assumptions

Given: A knot diagram and its two possible orientations.

[F1]

For a diagram with $c$ components, the $2^{c}$ canonical orientation cycles form a basis of Lee homology (Lee filtration, spectral sequence, and orientation classes).

[F2]

Filtered Lee homology is independent of the diagram of the oriented link (Lee filtered homology is an oriented link invariant).

[A1]

${A C}_{ω}$ is the countable-choice principle used by the finite diagram-move supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

Given: ${A C}_{ω}$ and a finite regular diagram $D$ of $K$ .

1.1F1givenalgebra

A knot has exactly one component, so it has exactly $2^{1} = 2$ orientations. Applying [F1] with $c = 1$ shows that the associated canonical classes are linearly independent and span $H (C_{L} (D; Q))$ . Thus its dimension is two.

2.1A1F2step 1.1∎

If $D^{'}$ is another diagram of the same oriented knot, [A1] supplies the axiom branch used in [F2], and [F2] gives an isomorphism of Lee homology preserving the induced filtration. Therefore the dimension and filtered vector-space type computed from $D$ agree with those computed from $D^{'}$ . The two canonical classes on each diagram form a basis by step 1.1; no identification of individual basis vectors across diagrams is needed for the claimed rank or filtration invariance.

Depends on

Used by

Rasmussen's invariant for positive knots Corollary
Rasmussen’s s-invariant Definition
Unknot, trefoils, and the 2-twisted positive Whitehead double Example
Lee cobordism maps give the genus filtration bound Lemma
Rasmussen's invariant changes sign under mirror image Lemma

Dependency tree · two levels

17 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)