Khovanov chain complex from a diagram

Input: a nonempty finite regular oriented link diagram $D$ and a total order $c_1<\cdots <c_n$ on its crossing set. A diagram object is supplied as mathematical input; this procedure does not parse a serialized string.

Output: the normalized bigraded Khovanov cochain module $C^{i,j}(D)$ and its candidate differential $d:C^{i,j}(D)\to C^{i+1,j}(D)$, constructed by the following finite instructions.

1. Enumerate every state $v:C(D)\to \{0,1\}$ once. Form the crossingless resolution by using the A-smoothing at $v(c)=0$ and the B-smoothing at $v(c)=1$.
2. For each state, create one tensor factor of the integral Frobenius algebra for each resolution circle and place the resulting tensor product in cohomological degree $\mathrm{\mid }v\mathrm{\mid }-n_{-}$ and quantum degree $\mathrm{\mid }v\mathrm{\mid }+n_{+}-2n_{-}+\sum \mathrm{deg}(a)$ for its circle labels $a$.
3. For each state and each crossing $c_j$ with $v(c_j)=0$, create the edge to $v+e_j$. Apply multiplication if that smoothing change merges two circles, comultiplication if it splits one circle, the identity on other tensor factors, and the sign $(-1{)}^{{\sum }_{i<j}v(c_i)}$. Sum the edge maps out of every state to obtain $d$.
4. Return the finite direct sum of the state modules with these degrees and maps. The square-zero property is the separate local lemma named by justified_by.

Regular diagram crossing set and A/B smoothing chart

Frobenius algebra, edge maps, and grading conventions

Khovanov, A categorification of the Jones polynomial (1999), §§3.3–4.2

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

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

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