Khovanov complexes are chain homotopy equivalent under Reidemeister moves

Crossing order and planar isotopy. Reordering adjacent crossing coordinates $a,b$ is corrected by the diagonal state factor $s(v)=(-1{)}^{v(a)v(b)}$: on an $a$-edge its ratio is $(-1{)}^{v(b)}$, and on a $b$-edge it is $(-1{)}^{v(a)}$, exactly the sign change from swapping their order. Thus local crossings can be ordered first, and can be restored to the original order afterward by explicit chain isomorphisms. A planar isotopy bijects labelled resolution circles and saddles, so its factor permutations give a grading-preserving chain isomorphism.

The determinant sign of each curl. A Reidemeister-I return joins one of the four adjacent endpoint pairs of [F7]. For the right-hand return choose the strand direction $SW\to NE\to SE\to NW$. At the crossing the over and under tangents are respectively $(1,1)$ and $(-1,1)$, whose determinant is $2>0$. A-smoothing closes the return $NE$--$SE$ into the small circle and leaves the left-hand continuing arc; B-smoothing has only the continuing arc. Thus this positive curl has the multiplication differential, small circle in local degree zero, used in the curl contraction. For the top return choose $SW\to NE\to NW\to SE$. The tangents are $(1,1)$ and $(1,-1)$, with determinant $-2<0$. B-smoothing closes the top return into the small circle while A-smoothing has only the continuing arc. Thus this negative curl has the comultiplication differential, small circle in local degree one. Rotation by $180$ degrees gives the left and bottom returns without changing A/B labels or determinant sign. Reversing the component direction negates both tangents and also leaves the determinant sign unchanged. These four return pairs and the two directions exhaust the oriented local R1 pictures; no orientation-reversing chart or reflection is used.

Both curls, with natural contractions. For a positive curl the local differential is multiplication from a small circle and a continuing arc. Put the small-circle factor first, denote the dot on the continuing arc by $x$, and use coordinates $1\otimes u+X\otimes v$. The map is $(u,v)\mapsto u+xv$. Its $1$-label component is the identity; cancel it, leaving $v\mapsto X\otimes v-1\otimes xv$ in degree zero. Equivalently this is $j=\mathrm{\Delta }-\mathrm{ȷ}m\mathrm{\Delta }$, with $\mathrm{ȷ}(v)=1\otimes v$; it sends $1$ to $X\otimes 1-1\otimes X$ and $X$ to $X\otimes X$. The canceled pair has inverse identity, so its contraction is the explicit cancellation rule of [F5], agreeing with the cochain identity-cone contraction in [F6]. For a continuing generator of quantum degree $d$, both terms of this survivor have raw quantum degree $d-1$ and local cohomological degree zero, so the raw quantum shift is $\{1\}$ in the stated lowering convention. For a negative curl the map is $v\mapsto (xv,v)$ into the circle coordinates; cancel its identity $X$-label component. The remaining $1$-label coordinate $1\otimes v$ has raw quantum degree $d+2$ (the circle label contributes $1$ and the local state contributes $1$), and is in local cohomological degree one. Thus its raw shifts are exactly $[-1]\{-2\}$. These descriptions commute with exterior saddles because the dot and the local saddle commute with them by the displayed Frobenius identities; they therefore contract the total complexes, including exterior differentials with their totalization signs. The additional normalizations are $\{-1\}$ for a positive curl and $[1]\{2\}$ for a negative curl, canceling the respective raw shifts.

The signed RII square. In the four states of the Remark take $U=E$, $V=I$, $W=E\bigsqcup ◯$, $Z=E$, after ordering the $(I,E)$ crossing first and the $(E,I)$ crossing second as in [F5]. That step gives the exact differential $u\mapsto (su,xu,u)$ and $(v,w_1,w_X)\mapsto tv-w_1-y{w}_X$, with $ts=x+y$, and the chain maps $i(v)=(v,tv,0)$, $p(v,w_1,w_X)=v-s{w}_X$, and homotopy $h^1=w_X$, $h^2(z)=(0,-z,0)$. Their checked identity is $1-ip=dh+hd$, $pi=1$. Thus the two canceled pairs are actual contractible summands over the integral exterior cube and the surviving identity matching is in local degree one with quantum increase one. This is precisely the raw shift $[-1]\{-1\}$ in the Statement. An RII pair has one positive and one negative crossing, so its normalization changes by $[1]\{1\}$ and cancels that shift. A quadratic tangency has two branches which are both graphs over their common tangent direction; orienting these graphs temporarily in the same direction puts every underlying RII picture into this model or the opposite crossing order. Reversing either actual strand arrow only changes crossing signs, still one of each, and does not change the A/B maps. Rotation of the local coordinate chart transports the same model; its crossing-order correction was proved in step 1.1.

The signed RIII model. Let $P_i$ denote the crossing complex with A-state $I$ and B-state $E_i$; this description fixes the smoothing convention without using a crossing-sign mnemonic. In [F5], both words $P_1{P}_2{P}_1$ and $P_2{P}_1{P}_2$ are retracted by two explicit cancellations to $I\stackrel{\left(\genfrac{}{}{0px}{}{s_1}{s_2}\right)}{\to }{E}_1\oplus E_2\stackrel{\left(\begin{array}{cc}-u& v\\ w& -z\end{array}\right)}{\to }{E}_1{E}_2\oplus E_2{E}_1\mathrm{.}$ Here $u$ appends $E_2$ below $E_1$, $v$ prepends $E_1$ above $E_2$, $w$ prepends $E_2$ above $E_1$, and $z$ appends $E_1$ below $E_2$; each is the actual ordered-tensor saddle specified there. The canceled maps are the $-1$ from the circle's $1$ coordinate in state $101$ to state $111$, and then the $+1$ from state $001$ to the circle's $X$ coordinate. The remaining cube signs are exactly the displayed matrix on both sides. Inclusion and projection after the coordinate changes of [F5] compose to mutually inverse chain maps up to the given contractions. These maps preserve quantum degree and commute with all exterior differentials by [F5]. This proves the raw, unshifted model RIII equivalence with explicit chain maps and homotopies.

All height and orientation variants. The opposite crossing complex $N_i$ has A-state $E_i$ and B-state $I$. Step 2.2 shows that $Q_i=N_i[1]\{1\}$ is a two-sided inverse of $P_i$ up to local chain homotopy; the contractions are natural under adjoining other crossings. Thus the relation $aba\simeq bab$, with $a=P_1,b=P_2$, also gives the relations with exponent triples $---$, $++-$, $-++$, $+--$ and $--+$, interpreting negative exponents by these explicit inverses. The $---$ relation is the inverse relation; $ab{a}^{-1}\simeq b^{-1}ab$ follows by multiplication on the left by $b^{-1}$ and on the right by $a^{-1}$; $a^{-1}ba\simeq ba{b}^{-1}$ follows by multiplication on the left by $a^{-1}$ and on the right by $b^{-1}$; their inverses give $+--$ and $--+$. Each multiplication is stacking a crossing complex and composing the actual chain maps and homotopies, so this is an operation on complexes, not an appeal to a braid-group invariant. A generic transverse triple event can be put in a small rectangle with all three branches monotone in one longitudinal coordinate: choose a direction transverse to their three tangents and use a sufficiently thin rectangle. Their two diagrams have braid words $121$ and $212$. The crossings concern the pairs of strands $AB,AC,BC$, respectively; a strict height order has precisely the six transitive sign patterns just listed, excluding the two cyclic patterns $+-+$ and $-+-$. For any such pattern the counts of $N_i$ agree on both sides, so the extra shifts in $Q_i$ cancel when removed. The actual oriented RIII move retains each of its three crossing signs; hence its normalizing $n_{+},n_{-}$ shifts also agree. Actual strand arrows may be arbitrary because the raw cube depends only on the smoothing chart. This proves every oriented RIII variant.

The based quotient and conclusion. Every type-I inclusion, projection and contraction in step 2.1 is a sum of circle-coordinate maps and dots away from the marked point. By the Frobenius formulas it commutes with $X_p$ even if its continuing arc lies on the distinguished resolution component. The same commutation for every RII/RIII map and homotopy, including compositions and inverse-pair cancellations, is proved in [F5]. Consequently all equations $pi=1$ and $1-ip=dh+hd$ descend modulo $\mathrm{im}{X}_p$. By [F4] this is exactly the reduced quotient, or equivalently the interval-module complex. Steps 1.1–3.1 and the checked shifts therefore prove every unreduced and based assertion of the Statement. All local constructions use finite integral formulas and are choice-free.