Lee filtered homology is an oriented link invariant

In $A_L=\mathbb{Q}[X]\mathrm{/}(X^2-1)$, multiplication by $X$ is a dot operator with $X^2=1$, and ${\mathrm{\Delta }}_L(z)=1\otimes Xz+X\otimes z,{\mathrm{\Delta }}_L{m}_L(a\otimes b)=Xa\otimes b+a\otimes Xb\mathrm{.}$ The first identity follows by substituting $z=1,X$; the second follows by substituting the four basis pairs $1\otimes 1,1\otimes X,X\otimes 1,X\otimes X$. Thus the split and merge circle coordinates in [F3] remain valid for the Lee algebra, and the neck-cutting relation required in the Reidemeister II retraction holds with $X^2=1$.

For a positive curl, the local differential is $(u,v)\mapsto u+Xv$ from the small-circle coordinates $1\otimes u+X\otimes v$ to the continuing arc. Change coordinates to $(u^{\mathrm{\prime }},v^{\mathrm{\prime }})=(u+Xv,v)$. The retraction has inclusion $i(w)=(0,w)$, projection $p(u^{\mathrm{\prime }},v^{\mathrm{\prime }})=v^{\mathrm{\prime }}$, and contraction $h(w)=(w,0)$ on the target; hence $pi=1$ and $1-ip=dh+hd$. For a negative curl, the local differential is $v\mapsto (Xv,v)$ into the small-circle coordinates. Change target coordinates to $(w_1^{\mathrm{\prime }},w_X)=(w_1-X{w}_X,w_X)$; then the map is $v\mapsto (0,v)$, and the retraction onto $w_1^{\mathrm{\prime }}$ has inclusion $i(w)=(w,0)$, projection $p(w_1^{\mathrm{\prime }},w_X)=w_1^{\mathrm{\prime }}$, and contraction $h(0,w)=w$ from the $w_X$ target summand to the source. These formulas give $pi=1$ and $1-ip=dh+hd$. Before the writhe shifts, the positive survivor has quantum shift $-1$ and the negative survivor has quantum shift $+2$; the new $+1$ and $-2$ writhe shifts cancel them, respectively. In both cases the normalized homological degree agrees with the original. For Reidemeister II use the local maps $i(v)=(v,tv,0)$ and $p(v,w_1,w_X)=v-s{w}_X$ from [F3]; the identity $ts=x+y$ used in $dh+hd=1-ip$ is exactly the second identity in step 1.1. Its local homological and quantum shifts are canceled by the one positive and one negative crossing normalization changes. For Reidemeister III, the two cancellations in [F4] use only identity pivots and the same split coordinates, while writhe normalization is unchanged. All coordinate changes have pivots $\pm 1$ and entries given by the Lee saddle and dot maps, so they preserve the decreasing filtration after normalization.

The Lee deformation terms raise $q$ by $4$ by [F1]. Each local coordinate change in step 2.1 is triangular with diagonal entries $\pm 1$; its off-diagonal entries are saddle maps, whose degree-preserving parts are the corresponding Khovanov maps and whose other terms raise $q$ by $4$. Therefore the maps, inverses, and contraction homotopies preserve the decreasing filtration after applying the normalization shifts. On the associated graded, the R1 retractions are the degree-preserving parts of the displayed curl maps, and the RII/RIII retractions are the local Khovanov equivalences of [F4]. Their filtered maps induce isomorphisms on filtered homology; their associated-graded maps induce isomorphisms on the Khovanov page and, by page iteration, on every later spectral-sequence page.

By [A1] and [F2], choose a finite sequence of oriented Reidemeister moves and planar isotopies from $D$ to $D^{\mathrm{\prime }}$; if $D=D^{\mathrm{\prime }}$, the empty sequence gives the identity. Planar isotopies rel crossing data simply relabel state circles and preserve all degrees. Compose the filtered equivalences from step 3.1 along this finite sequence. The composite has a filtered inverse up to filtered homotopy, so [F5] gives an isomorphism of filtered Lee homology and step 3.1 gives isomorphisms of all pages from rational Khovanov onward. This proves diagram independence. The only choice used is the finite move sequence supplied by ${\mathrm{A}\mathrm{C}}_{\omega }$; the local algebraic maps themselves are explicit and choice-free.