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Lee-deformed Khovanov complex and filtration

Definition

Let $D$ be a finite regular oriented link diagram in the standard oriented $3$ -sphere. Use the oriented Khovanov resolution cube, crossing order, signs, and shifts fixed in the integral Khovanov construction. Over $Q$ , replace its circle algebra by $A_{L} = Q [X] / (X^{2} - 1)$ , with $\deg_{q} (1) = 1$ and $\deg_{q} (X) = - 1$ , and structure maps

η_{L} (1) = 1, ϵ_{L} (1) = 0, ϵ_{L} (X) = 1,

m_{L} (1 \otimes 1) = 1, m_{L} (1 \otimes X) = m_{L} (X \otimes 1) = X, m_{L} (X \otimes X) = 1,

Δ_{L} (1) = 1 \otimes X + X \otimes 1, Δ_{L} (X) = 1 \otimes 1 + X \otimes X .

For a resolution state $v$ with $k (v)$ circles, assign $A_{L}^{\otimes k (v)}$ and cohomological degree $i (v) = ∣ v ∣ - n_{-}$ . On a tensor basis element whose circle labels are $a_{1}, \dots, a_{k (v)} \in {1, X}$ , set

q (v; a_{1}, \dots, a_{k (v)}) = ∣ v ∣ + n_{+} - 2 n_{-} + \sum_{j = 1}^{k (v)} \deg_{q} (a_{j}) .

The differential is the signed sum of the edge maps $m_{L}$ and $Δ_{L}$ . The resulting finite cochain complex is the Lee-deformed Khovanov complex $C_{L} (D; Q)$ . Its decreasing quantum filtration $F^{p} C_{L} (D; Q)$ is spanned by basis elements of quantum degree at least $p$ . Its homology with the induced filtration is the Lee homology of the diagram. The diagram-level definition does not by itself assert invariance under changing $D$ .

For the empty link diagram, use its single empty resolution state and empty tensor term $Q$ in cohomological and quantum degree $0$ , with zero differential. Its induced decreasing filtration is $F^{p} C_{L} (\emptyset; Q) = Q$ for $p \leq 0$ and $F^{p} C_{L} (\emptyset; Q) = 0$ for $p > 0$ .

The two equal composites in the diagram below are the Frobenius compatibility maps used on merge-split faces. The drawing names maps only; the proof checks the equality on the basis ${1, X}^{\otimes 2}$ .

Definition references

The crossing order, signed resolution cube, and quantum shift being deformed

The cochain-complex convention

Filtered-complex convention

Increasing and decreasing filtrations

Rasmussen, Khovanov homology and the slice genus (2004), §§2.3–2.4

McMullen, 4-Manifolds, §§5.5–5.6

Lee, On Khovanov invariant for alternating links (2002), §§2–3

Facts & Assumptions

Given: A finite regular oriented diagram $D$ , an ordering of its crossing set, and the coefficient field $Q$ .

[F1]

The normalized Khovanov cube uses a finite state set, the state degree $∣ v ∣ - n_{-}$ , the quantum shift $∣ v ∣ + n_{+} - 2 n_{-}$ , and a cube sign that reverses on opposite routes around each square face. (Khovanov Frobenius algebra and cube of resolutions)

[F2]

A cochain complex has a degree- $+ 1$ differential whose square is zero. (Cochain complex in an abelian category)

Proof

Given: The finite oriented resolution cube of $D$ and the operations in the Definition.

Empty-link case: For $D = \emptyset$ , the state set is the singleton empty function, its empty tensor term is $Q$ in degrees $(0, 0)$ , and there are no edges. The differential is zero and the degree-zero generator gives exactly the filtration stated in the Definition. Assume henceforth that $D$ is nonempty for the remaining checks.

1.1givenalgebra

The quotient $A_{L} = Q [X] / (X^{2} - 1)$ has basis $1, X$ , and its multiplication table is the displayed $m_{L}$ . Hence multiplication is associative and commutative with unit $1$ . The counit identities follow on $1, X$ : $(ϵ_{L} \otimes i d) Δ_{L} (1) = 1$ , $(ϵ_{L} \otimes i d) Δ_{L} (X) = X$ , and the other counit composite gives the same values.

2.1givenstep 1.1algebra

For input $1$ , both composites $(Δ_{L} \otimes i d) Δ_{L}$ and $(i d \otimes Δ_{L}) Δ_{L}$ equal $1 \otimes X \otimes X + X \otimes 1 \otimes X + 1 \otimes 1 \otimes 1 + X \otimes X \otimes 1$ . For input $X$ , both equal $1 \otimes 1 \otimes X + 1 \otimes X \otimes 1 + X \otimes 1 \otimes 1 + X \otimes X \otimes X$ . Thus $Δ_{L}$ is coassociative. Cocommutativity follows by swapping tensor factors in its two displayed values.

3.1F1F2step 1.1step 2.1algebra

On the ordered basis $1 \otimes 1, 1 \otimes X, X \otimes 1, X \otimes X$ , each of $(m_{L} \otimes i d) (i d \otimes Δ_{L})$ , $Δ_{L} m_{L}$ , and $(i d \otimes m_{L}) (Δ_{L} \otimes i d)$ has outputs $1 \otimes X + X \otimes 1$ , $1 \otimes 1 + X \otimes X$ , $1 \otimes 1 + X \otimes X$ , and $1 \otimes X + X \otimes 1$ . The Frobenius identities therefore hold. On a cube square, two changes on disjoint circles commute by tensor functoriality; two merges give the same composite by associativity of $m_{L}$ ; two splits give the same composite by coassociativity of $Δ_{L}$ ; and a merge-split pair gives the same composite by the Frobenius identities just checked. In each case the cube signs on the two routes are opposite, so the square contribution to $d_{L}^{2}$ is zero. Every length-two path in the cube lies in one such square, so summing over squares proves $d_{L}^{2} = 0$ .

4.1F1F2step 3.1algebra∎

The terms of $m_{L}$ and $Δ_{L}$ inherited from $Q [X] / (X^{2})$ have total quantum degree $0$ after the state shift. The new multiplication term $m_{L} (X \otimes X) = 1$ changes the label degree from $- 2$ to $1$ , and the edge state shift adds $1$ , so it raises $q$ by $4$ . The new term $1 \otimes 1$ in $Δ_{L} (X)$ changes label degree from $- 1$ to $2$ , and its edge state shift again adds $1$ , so it raises $q$ by $4$ . Thus every differential term preserves or raises $q$ , making each $F^{p} C_{L}$ a subcomplex. The state set and each state module are finite, so only finitely many $q$ -degrees occur and the filtration is finite. All algebra and cube checks use finite bases and no choice axiom.

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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)
Eun Soo Lee, On Khovanov invariant for alternating links, arXiv:math/0210213v2 (standard reference, not scraped)