Reduced Khovanov cohomology of a based link over F2

If an edge avoids the marked circle, its map acts as the identity on that factor and commutes with $X_p$. For a merge involving the marked circle, associativity gives $m(Xa\otimes b)=Xm(a\otimes b)$, with the marked factor placed on the appropriate input. For a split of that circle, the Frobenius identity gives $\mathrm{\Delta }(Xa)=(X\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }(a)=(\mathrm{i}\mathrm{d}\otimes X)\mathrm{\Delta }(a)$; for $a=1$ both sides equal $X\otimes X$, and for $a=X$ both sides vanish. Thus every edge map commutes with $X_p$, so $\mathrm{im}{X}_p$ is a subcomplex.

The quotient differential is well-defined by step 1.1 and squares to zero by [F3]. In each state the marked-circle factor has basis $1,X$, and multiplication by $X$ has image the one-dimensional span of $X$. The quotient therefore replaces that factor by ${\mathbb{F}}_2\{1\}$; the shift $j-1$ assigns the surviving label degree zero. This gives the stated reduced cochain complex and its cohomology.