The E1 differential of the link-surgery complex is reduced Khovanov

The filtration quotient at each cube vertex is its resolution Floer group, and the differential on the first page is the sum of components changing one crossing.

Applying the asserted edge identifications [F2] to each of the finitely many edges identifies this first-page differential with the reduced Khovanov cube differential; the finite direct sum of the vertex isomorphisms is an isomorphism of cochain complexes.

Write $n_{\pm }=n_{\pm }(D)$ and let $a_c\in \{1,X\}$ label the unmarked circles of a state $I$, with quantum degrees $1,-1$. Mirroring interchanges $n_{+}$ and $n_{-}$, so the normalized target convention in [F2] gives $i=\mathrm{\mid }I\mathrm{\mid }-n_{+}$ and $\tilde{j}={\sum }_{c\mathrm{\ne }p}{\mathrm{deg}}_q(a_c)+\mathrm{\mid }I\mathrm{\mid }+n_{-}-2n_{+}$. The marked quotient label has reduced degree zero. Unmarked saddles lower the label-degree sum by one. A marked merge sends the other circle's $1$ to the marked $1$ and kills its $X$; a marked split creates $X$ on the new unmarked circle. Their nonzero terms also lower the unmarked degree sum by one. Thus every edge raises $i$ by one and preserves $\tilde{j}$. These degrees are transported from the reduced state bases to $E_1$; no independent Floer quantum grading is asserted. For filtration index $s=n-\mathrm{\mid }I\mathrm{\mid }$ the cochain reindexing is $i=n_{-}-s$.

Taking cohomology of the identified first-page complexes gives the based reduced Khovanov cohomology on the second page, with the shifts checked in step 3.1. By [F3] its graded isomorphism class is independent of the chosen link component for this fixed diagram. The actual Floer saddle and mirror identifications are the supplier inputs in [F2]; the finite assembly and normalization above introduce no further geometric assertion. This proves the stated identification and asserts no map between distinct diagram presentations.