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Khovanov Frobenius algebra and cube of resolutions

Definition

Let $D$ be a nonempty finite regular oriented link diagram in an oriented $3$ -sphere, with finite crossing set $C (D)$ . Fix a unit viewing normal $n$ and orient the projection plane $P$ so $(u, v)$ is positive exactly when $(n, u, v)$ is positive in the ambient $3$ -space. At a crossing $c$ , define its sign as the sign of $\det_{P} (v_{o v e r}, v_{u n d e r})$ , where the two nonzero projected tangent vectors follow the oriented over- and under-strands. Let $n_{+}$ and $n_{-}$ be the numbers of positive and negative crossings.

Set $A = Z [X] / (X^{2})$ , with $\deg (1) = 1$ and $\deg (X) = - 1$ . Its unit, multiplication, counit, and comultiplication are

η (1) = 1, m (1 \otimes 1) = 1, m (1 \otimes X) = m (X \otimes 1) = X, m (X \otimes X) = 0,

ϵ (1) = 0, ϵ (X) = 1, Δ (1) = 1 \otimes X + X \otimes 1, Δ (X) = X \otimes X .

Thus $m$ and $Δ$ have quantum degree $- 1$ , while $η$ and $ϵ$ have degree $+ 1$ .

Choose a total order $c_{1} < \dots < c_{n}$ on $C (D)$ . A resolution state is a function $v : C (D) \to {0, 1}$ ; at $v (c) = 0$ use the A-smoothing and at $v (c) = 1$ use the B-smoothing from the Kauffman source chart. Write $∣ v ∣ = \sum_{c} v (c)$ , let $D_{v}$ be the crossingless resolution, and let $k (v)$ be its number of circles. Assign the state module $A^{\otimes k (v)}$ . For an edge $v \to v + e_{j}$ changing crossing $c_{j}$ from $0$ to $1$ , use $m$ if the local smoothing merges two circles and $Δ$ if it splits one circle, tensored with the identity on all other factors, and multiply this map by $(- 1)^{\sum_{i < j} v (c_{i})}$ . The resulting direct sum has cohomological degree $i (v) = ∣ v ∣ - n_{-}$ and the generator with circle labels $1$ or $X$ has quantum degree $j (v) = ∣ v ∣ + n_{+} - 2 n_{-} + \sum_{circle labels a} \deg (a) .$ This defines the normalized integral Khovanov cube and its candidate degree-zero cochain differential; the assertion that the differential squares to zero is a separate lemma.

The Frobenius identity that controls a merge-split face is represented by the two equal composites below. The diagram records the maps only; the proof verifies the equality on the basis ${1, X}^{\otimes 2}$ and does not rely on the drawing.

Definition references

Kauffman smoothing chart and finite crossing set

Determinant crossing sign and writhe

Cochain-complex convention

Khovanov, A categorification of the Jones polynomial (1999), §§2.2–2.3 and 4.2

Bar-Natan, On Khovanov's categorification of the Jones polynomial (2002), §§2–3

Kauffman, An Introduction to Khovanov Homology (2022), §§3–4

Zhang, Notes on Khovanov homology (2025), §§3.5–3.7

Facts & Assumptions

Given: A nonempty finite regular oriented diagram $D$ , a total order of its crossing set, and the operations displayed in the Definition.

Proof

1.1givenalgebra

The quotient by the monic polynomial $X^{2}$ is the free abelian group with basis $1, X$ . The displayed multiplication is the quotient-ring multiplication, so it is associative, commutative, and has unit $1$ . On this basis, $ϵ$ and $Δ$ satisfy both counit identities; explicitly, $(ϵ \otimes i d) Δ (1) = 1$ , $(i d \otimes ϵ) Δ (1) = 1$ , and the same composites send $X$ to $X$ .

1.2givenalgebra

Cocommutativity follows by swapping tensor factors in the two displayed values of $Δ$ . For coassociativity, both $(Δ \otimes i d) Δ (1)$ and $(i d \otimes Δ) Δ (1)$ equal $1 \otimes X \otimes X + X \otimes 1 \otimes X + X \otimes X \otimes 1$ , and both composites send $X$ to $X \otimes X \otimes X$ . Hence $(A, m, η, Δ, ϵ)$ is a commutative Frobenius algebra once the Frobenius compatibility is checked.

2.1givenstep 1.1step 1.2algebra∎

On the ordered basis $1 \otimes 1, 1 \otimes X, X \otimes 1, X \otimes X$ of $A \otimes A$ , each of $(m \otimes i d) (i d \otimes Δ)$ , $Δ m$ , and $(i d \otimes m) (Δ \otimes i d)$ has respective outputs $1 \otimes X + X \otimes 1$ , $X \otimes X$ , $X \otimes X$ , and $0$ . Thus the Frobenius identities hold. The degree shifts in the Definition give each edge map total quantum degree $0$ ; the quotient-ring construction and all checks are choice-free.

Depends on

Used by

Khovanov chain complex from a diagram Algorithm
Rasmussen's invariant for positive knots Corollary
A birth and a death do not cancel Counterexample
Omitting Khovanov normalization breaks Reidemeister I and the Jones Euler characteristic Counterexample
Lee-deformed Khovanov complex and filtration Definition
Reduced Khovanov cohomology of a based link over F2 Definition
Reduced Khovanov cohomology of a knot Definition
The Khovanov map assigned to a generic surface movie Definition
A birth followed by a merging saddle Example
Reduced Khovanov ranks of the unknot and the three-crossing two-braid Example
Unknot Khovanov and reduced Khovanov groups Example
Filtered instanton complex from the cube of knot resolutions Lemma
Khovanov complexes are chain homotopy equivalent under Reidemeister moves Lemma
Odd Khovanov and Bar–Natan LEO foundations Lemma
Reduced Khovanov homology over F2 is independent of the chosen link component Lemma
Signed local cancellation models for the Khovanov Reidemeister moves Lemma
The graded Euler characteristics have the Jones normalization Lemma
The Khovanov cube differential squares to zero Lemma
The Khovanov Frobenius TQFT respects the Bar-Natan relations Lemma
The reduced Khovanov homology of a trefoil has rank three Lemma

Dependency tree · two levels

12 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

Mikhail Khovanov, A categorification of the Jones polynomial, arXiv:math/9908171v2, §§2.2–2.3 and 4.2 (standard reference, not scraped)
Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebraic & Geometric Topology 2 (2002), §§2–3 (standard reference, not scraped)
Louis H. Kauffman, An Introduction to Khovanov Homology, arXiv:1107.1524v2 (2022), §§3–4 (standard reference, not scraped)
Melissa Zhang, Notes on Khovanov homology, arXiv:2501.03115v1, §§3.5–3.7 (standard reference, not scraped)