Khovanov's generator complexes for the HHH construction

In the reduced setting of The reduced type-A polynomial ring and Soergel bimodules for the HHH construction define, for $1\le i\le m-1$, degree-zero maps of graded $(R,R)$-bimodules $b{r}_i\text{\hspace{0.17em}⁣}:B_i\to R,b{r}_i(a\otimes b)=ab(b{r}_i(1\otimes 1)=1),$ $r{b}_i\text{\hspace{0.17em}⁣}:R\{2\}\to B_i,r{b}_i(1)=y_i\otimes 1+1\otimes y_i,y_i=x_i-x_{i+1}\mathrm{.}$ Khovanov's generator complexes are the bounded complexes of graded $(R,R)$-bimodules with degree-zero differentials $F({\sigma }_i):=[R\{2\}\stackrel{r{b}_i}{\to }{B}_i],F({\sigma }_i^{-1}):=[B_i\{-2\}\stackrel{b{r}_i}{\to }R\{-2\}],$ where in $F({\sigma }_i)$ the term $R\{2\}$ sits in cohomological degree $-1$ and the term $B_i$ in degree $0$, and in $F({\sigma }_i^{-1})$ the term $B_i\{-2\}$ sits in cohomological degree $0$ and the term $R\{-2\}$ in degree $1$. For a signed word $\sigma ={\sigma }_{i_1}^{{ϵ}_1}\cdots {\sigma }_{i_r}^{{ϵ}_r}$ put $F(\sigma ):=F({\sigma }_{i_1}^{{ϵ}_1}){\otimes }_R\cdots {\otimes }_{R}F({\sigma }_{i_r}^{{ϵ}_r}),$ the signed tensor totalization of Bounded graded bimodule complexes and signed tensor totalization, with the empty word giving the unit complex $R$; it is a bounded complex of graded $(R,R)$-bimodules with degree-zero differentials, whose terms carry a cohomological and an internal grading.

Normalization comparison with the library's Rouquier complexes. The library's The positive and negative Rouquier generator complexes is stated for the ambient polynomial ring $\mathbb{Q}[x_1,\dots ,x_m]$; its reduced instance is obtained by replacing that ring by the reduced ring $R$ of The reduced type-A polynomial ring and Soergel bimodules for the HHH construction and $R^{s_i}$ by the invariant subring. In that instance the shifted bimodule is $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}=R{\otimes }_{R^{s_i}}R(1)=B_i\{-1\},\text{so}{F}_i=[B_i\{-1\}\stackrel{{\epsilon }_i}{\to }R\{-1\}],F_i^{-1}=[R\{1\}\stackrel{{\eta }_i}{\to }{B}_i\{-1\}],$ with the $B$-term in cohomological degree $0$ in both complexes and ${\eta }_i(1)={\alpha }_i\otimes 1+1\otimes {\alpha }_i$, where ${\alpha }_i$ is the balanced root. Letter by letter, $F({\sigma }_i)\cong F_i^{-1}\{1\},F({\sigma }_i^{-1})\cong F_i\{-1\}$ in the library's $\{r\}$ notation: the shift $\{1\}$ turns $R\{1\}$ into $R\{2\}$ and $B_i\{-1\}$ into $B_i$, and the balanced root satisfies ${\alpha }_i={\kappa }_i{y}_i$ with ${\kappa }_i=\pm 1$ a unit, so $r{b}_i$ and ${\eta }_i$ differ by the unit ${\kappa }_i$; the same computation, with multiplication in both differentials, matches $F({\sigma }_i^{-1})$ with $F_i\{-1\}$. Hence for a word $\sigma$ the complex $F(\sigma )$ is, up to the overall internal shift $\{{ϵ}_1+\cdots +{ϵ}_r\}$ (the writhe), the Rouquier complex The Rouquier complex of a braid word of the generator-inverted word ${\sigma }_{i_1}^{-{ϵ}_1}\cdots {\sigma }_{i_r}^{-{ϵ}_r}$; by The Rouquier complex is well defined up to canonical homotopy equivalence it is determined by the underlying braid up to canonical homotopy equivalence and that shift.

Caveats. The pairing of generator and complex (the $R$-term below the $B$-term for ${\sigma }_i$) is the one matched by the positive-crossing cone of The positive and negative Khovanov-Rozansky crossing complexes; the comparison with the library's complexes is an isomorphism in the homotopy category, not an equality of complexes; the direction of the word comparison (generator inversion) is forced by the opposite pairing used in The positive and negative Rouquier generator complexes; and the reduced instance inherits the ambient homotopy comparisons by the polynomial-extension and specialization argument in step 3.1.