Hochschild Homology and Triply-Graded Link Homology

Fix $m\ge 1$ and let $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$ be the polynomial ring of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution in $m$ commuting indeterminates, with the place-permutation action of the symmetric group $S_m$, the grading $\mathrm{deg}{x}_i=2$, the adjacent transpositions $s_i=(ii+1)$ and the invariant rings $(R^{\mathrm{\prime }}{)}^{s_i}$ of The standard type-A reflection realization and its polynomial ring. Write $x_i$ for the $i$-th coordinate function and let $S_m$ act by $w\cdot x_i=x_{w(i)}$.

The reduced ring. Put $y_i:=x_i-x_{i+1}$ for $1\le i\le m-1$ and $R:=\mathbb{Q}[y_1,\dots ,y_{m-1}]=\mathbb{Q}[x_1-x_2,x_2-x_3,\dots ,x_{m-1}-x_m]\subset R^{\mathrm{\prime }},$ the reduced polynomial ring of the HHH construction, with the restricted $S_m$-action and the induced grading $\mathrm{deg}{y}_i=2$. On the displayed generators the restricted action is $s_i(y_i)=-y_i,s_i(y_{i+1})=y_i+y_{i+1},s_i(y_{i-1})=y_{i-1}+y_i,s_i(y_j)=y_j(\mathrm{\mid }i-j\mathrm{\mid }\ge 2),$ and it extends uniquely to a $\mathbb{Q}$-algebra automorphism of $R$, because the displayed polynomials lie in $R$. Write $R^{s_i}\subset R$ for the invariant subring of the involution $s_i$; put $z_j=y_j+\frac{1}{2}{y}_i$ when $\mathrm{\mid }j-i\mathrm{\mid }=1$ and $z_j=y_j$ when $\mathrm{\mid }j-i\mathrm{\mid }\ge 2$. Then $s_i$ fixes each $z_j$ and sends $y_i$ to $-y_i$; the linear change of variables is invertible, and explicitly $R^{s_i}=\mathbb{Q}[z_j(j\mathrm{\ne }i),y_i^2]$.

The invariant coordinate. For $1\le i\le m-1$ put $t_i:=x_i+x_{i+1}\in R^{\mathrm{\prime }}$. Then $s_i$ fixes $t_i$, and the substitution $x_i=\frac{1}{2}(t_i+y_i),x_{i+1}=\frac{1}{2}(t_i-y_i),x_{i+2}=x_{i+1}-y_{i+1},x_{i-1}=x_i+y_{i-1},\dots$ expresses every coordinate as a polynomial in $t_i$ and the $y_j$ with coefficients in $\mathbb{Q}$ ($2$ is invertible). The substitution $\phi \text{\hspace{0.17em}⁣}:\mathbb{Q}[y_1,\dots ,y_{m-1},T]\to R^{\mathrm{\prime }}$ with $y_j\mapsto y_j$ and $T\mapsto t_i$ is the linear change of variables $(x_1,\dots ,x_m)\leftrightarrow (y_1,\dots ,y_{m-1},t_i)$, which is invertible over $\mathbb{Q}$ by the preceding display, hence an isomorphism of graded rings onto $R^{\mathrm{\prime }}$; consequently $R^{\mathrm{\prime }}=R[t_i]=R{\otimes }_{\mathbb{Q}}\mathbb{Q}[t_i]$ as graded rings. This polynomial extension is a free graded $R$-module on the infinite basis $1,t_i,t_i^2,\dots$. The extension of the invariant subring instead has rank two: $R^{\mathrm{\prime }}$ is free over $(R^{\mathrm{\prime }}{)}^{s_i}$ on $1,y_i$ (or on $1,x_i$), since $R=R^{s_i}\oplus y_i{R}^{s_i}$. Since $s_i$ fixes $t_i$ and acts on the coefficients through $R$, $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[t_i]=R^{s_i}{\otimes }_{\mathbb{Q}}\mathbb{Q}[t_i]\mathrm{.}$ Concretely $(R^{\mathrm{\prime }}{)}^{s_i}=\mathbb{Q}[x_1,\dots ,x_{i-1},t_i,x_i{x}_{i+1},x_{i+2},\dots ,x_m]$ and $x_i{x}_{i+1}=\frac{1}{4}(t_i^2-y_i^2)$; both displays generate the same subring because $y_i^2=t_i^2-4x_i{x}_{i+1}$.

The reduced simple-reflection bimodule. Since $2$ is invertible in $\mathbb{Q}$, the averaging idempotent $\frac{1}{2}(1+s_i)$ splits $R=R^{s_i}\oplus y_i{R}^{s_i}$ as graded $R^{s_i}$-modules: for $f\in R$ the element $\frac{1}{2}(f-s_i(f))$ equals $y_{i}g$ with $g=\frac{1}{2}(f-s_i(f))\mathrm{/}{y}_i\in R^{s_i}$. Hence the balanced tensor product of the graded $(R,R^{s_i})$-bimodule $R$ with the graded $(R^{s_i},R)$-bimodule $R$ $B_i:=R{\otimes }_{R^{s_i}}R$ is free of rank two as a left $R$-module and as a right $R$-module, with left basis $1\otimes 1,1\otimes y_i$ and right basis $1\otimes 1,y_i\otimes 1$, of degrees $0,2$ on either side; we use the library's internal shift convention $(M\{r\}{)}_d=M_{d-r}$ for graded modules, in which $B_i$ carries no internal shift.

Comparison with the published type-A bimodule. The published type-A Soergel bimodule of a simple reflection works with the ambient ring $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$ in place of the reduced ring and with the shifted bimodule $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}:=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}(1)=B_i^{\mathrm{\prime }}\{-1\},$ where $B_i^{\mathrm{\prime }}:=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}$ is the corresponding unshifted balanced tensor; the dictionary is extended on the next item of this page. In particular the element $1\otimes 1$ of the published bimodule has degree $-1$, while the element $1\otimes 1$ of $B_i$ has degree $0$.

Source convention. Khovanov writes $R^{\mathrm{\prime }}=R{\otimes }_{\mathbb{Q}}\mathbb{Q}[x_j]$ for the coordinate $x_j$ and chooses $j=1$. Read with $x_1$ this is exact as a statement about graded rings, but the identification of invariant subrings $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[x_1]$ is a literal equality of subrings of $R^{\mathrm{\prime }}$ only when $s_i$ fixes $x_1$, that is for $i\ge 2$; for $i=1$ the $s_1$-invariant coordinate is $t_1=x_1+x_2$, and all statements of this page are therefore stated with the invariant coordinate $t_i$, the two forms being related by the substitution above. This is a convention correction, not a change of the source's construction.

Small cases. For $m=1$ there are no differences and $R=\mathbb{Q}$; there are no simple reflections, and the empty-word bimodule is $R$ itself. For $m=2$ one has $R=\mathbb{Q}[y_1]$, $R^{s_1}=\mathbb{Q}[y_1^2]$ and $B_1=\mathbb{Q}[y_1]{\otimes }_{\mathbb{Q}[y_1^2]}\mathbb{Q}[y_1]$.

No choice principle is used in this definition.

Keep the ambient ring $R^{\mathrm{\prime }}=\mathbb{Q}[x_1,\dots ,x_m]$, the reduced ring $R\subset R^{\mathrm{\prime }}$ generated by the differences $y_j=x_j-x_{j+1}$, the invariant rings $R^{s_i}$, the invariant coordinates $t_i=x_i+x_{i+1}$ and the reduced bimodules $B_i=R{\otimes }_{R^{s_i}}R$ of The reduced type-A polynomial ring and Soergel bimodules for the HHH construction, with the internal shift convention $(M\{r\}{)}_d=M_{d-r}$ of Associative graded algebras, bimodules, and internal shifts.

The unreduced simple-reflection bimodule. For a simple reflection $s_i$ put $R_i^{\mathrm{\prime }}:=\mathbb{Q}[x_1,\dots ,x_{i-1},x_i+x_{i+1},x_i{x}_{i+1},x_{i+2},\dots ,x_m]=(R^{\mathrm{\prime }}{)}^{s_i},$ the $s_i$-invariant subring of $R^{\mathrm{\prime }}$ (symmetric polynomials in $x_i,x_{i+1}$), and $B_i^{\mathrm{\prime }}:=R^{\mathrm{\prime }}{\otimes }_{R_i^{\mathrm{\prime }}}{R}^{\mathrm{\prime }}$ with no internal shift. Since $R^{\mathrm{\prime }}=R[t_i]$ and $(R^{\mathrm{\prime }}{)}^{s_i}=R^{s_i}[t_i]$ by The reduced type-A polynomial ring and Soergel bimodules for the HHH construction, and $R^{\mathrm{\prime }}=R^{s_i}[t_i]\oplus y_i{R}^{s_i}[t_i]$, the bimodule $B_i^{\mathrm{\prime }}$ is free of rank two as a left and as a right $R^{\mathrm{\prime }}$-module, with left basis $1\otimes 1,1\otimes y_i$ and right basis $1\otimes 1,y_i\otimes 1$, of degrees $0,2$ on either side.

The trivial polynomial factor. Write $t=t_i$ and let $R_t\subseteq R^{\mathrm{\prime }}$ denote the polynomial ring $\mathbb{Q}[t]$ regarded as a subring of $R^{\mathrm{\prime }}$; the coordinate $t$ is $s_i$-invariant, hence central. The assignment on $R_i^{\mathrm{\prime }}$-balanced tensors $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:B_i^{\mathrm{\prime }}⟶B_i{\otimes }_{\mathbb{Q}}\mathbb{Q}[t],\mathrm{\Phi }(r{t}^a\otimes r^{\mathrm{\prime }}{t}^b)=(r\otimes r^{\mathrm{\prime }})\otimes t^{a+b}(r,r^{\mathrm{\prime }}\in R,a,b\ge 0)$ is an isomorphism of graded $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodules, where $R^{\mathrm{\prime }}=R[t]$ acts on the target by $R$ on the $B_i$-factor and by multiplication on $\mathbb{Q}[t]$. It is well defined: if $h={\sum }_a{h}_a{t}^a\in R^{s_i}[t]=R_i^{\mathrm{\prime }}$ with $h_a\in R^{s_i}$, then $\mathrm{\Phi }(h{t}^c\otimes 1)={\sum }_a(h_a\otimes 1)\otimes t^{a+c}$ and $\mathrm{\Phi }(1\otimes h{t}^c)={\sum }_a(1\otimes h_a)\otimes t^{a+c}$ agree because $h_a\otimes 1=1\otimes h_a$ in $B_i$. It is bijective because both sides are free of rank two over $R^{\mathrm{\prime }}$ on the respective generators $1\otimes 1,1\otimes y_i$ and $(1\otimes 1)\otimes 1,(1\otimes y_i)\otimes 1$, which $\mathrm{\Phi }$ matches. The factor $\mathbb{Q}[t]$ is the trivial polynomial factor: the coordinate $t$ acts by the same multiplication on both sides of the target, which is exactly the statement that it lies in the invariant ring $(R^{\mathrm{\prime }}{)}^{s_i}$ and hence balances in $B_i^{\mathrm{\prime }}$.

The reduced maps and their trivial extensions. Define degree-zero maps of graded $(R,R)$-bimodules $b{r}_i\text{\hspace{0.17em}⁣}:B_i\to R,b{r}_i(a\otimes b)=ab,$ $r{b}_i\text{\hspace{0.17em}⁣}:R\{2\}\to B_i,r{b}_i(1)=y_i\otimes 1+1\otimes y_i\mathrm{.}$ Both are well defined: $b{r}_i$ kills the balancing relation $h\otimes 1-1\otimes h$ for $h\in R^{s_i}$, and $r{b}_i$ is left $R$-linear by construction while the identity $r(y_i\otimes 1+1\otimes y_i)=(y_i\otimes 1+1\otimes y_i)r$ holds for every invariant generator $z_j$ because it balances, and for $r=y_i$ because $y_i^2\in R^{s_i}$. These elements generate $R$, as proved in The reduced type-A polynomial ring and Soergel bimodules for the HHH construction. Both maps have degree zero in the internal grading, since $\mathrm{deg}{y}_i=2$ and $\mathrm{deg}1\in R\{2\}=2$. Likewise $b{r}_i^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:B_i^{\mathrm{\prime }}\to R^{\mathrm{\prime }},b{r}_i^{\mathrm{\prime }}(a\otimes b)=ab,$ $r{b}_i^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:R^{\mathrm{\prime }}\{2\}\to B_i^{\mathrm{\prime }},r{b}_i^{\mathrm{\prime }}(1)=(x_i-x_{i+1})\otimes 1+1\otimes (x_i-x_{i+1}),$ are well-defined degree-zero $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodule maps by the same argument with $R^{\mathrm{\prime }}$ in place of $R$ and $(R^{\mathrm{\prime }}{)}^{s_i}$ in place of $R^{s_i}$. Under $\mathrm{\Phi }$ the elements $1\otimes 1\mapsto (1\otimes 1)\otimes 1$ and $y_i\otimes 1+1\otimes y_i\mapsto r{b}_i(1)\otimes 1$ correspond, so $b{r}_i^{\mathrm{\prime }}$ and $r{b}_i^{\mathrm{\prime }}$ restrict to the trivial-factor extensions of $b{r}_i$ and $r{b}_i$.

Products over the layers. Let $D$ be a marked MOY resolution with $r$ wide edges, at positions $s_1,\dots ,s_r$ in the layer order, and put $B^{\mathrm{\prime }}(D):={⨂}_{j=1}^r{B}_{s_j}^{\mathrm{\prime }},B(D):={⨂}_{j=1}^r{B}_{s_j},$ the balanced tensor products over $R^{\mathrm{\prime }}$ respectively over $R$ in the layer order. Choose the common invariant coordinate $t=(x_1+\cdots +x_m)\mathrm{/}m$; then $R^{\mathrm{\prime }}=R[t]$ and $t$ is fixed by every simple reflection. Applying the preceding single-factor construction with this $t$ gives an isomorphism of graded $(R^{\mathrm{\prime }},R^{\mathrm{\prime }})$-bimodules $B^{\mathrm{\prime }}(D)⟶B(D){\otimes }_{\mathbb{Q}}\mathbb{Q}[t]\mathrm{.}$ Indeed, each factor is $B_{s_j}\otimes \mathbb{Q}[t]$, and balancing over $R[t]$ combines the polynomial factors by multiplication. The inverse sends $(b_1{\otimes }_R\cdots {\otimes }_R{b}_r)\otimes t^a$ to the tensor with $t^a$ in its first factor. For $r=0$ this is the ring identity $R^{\mathrm{\prime }}=R[t]$. The two maps preserve the left and right actions and grading. An arbitrary coordinate such as $x_1$ need not act equally on both sides; its left and right actions are recovered from $x_1=t+(x_1-t)$, with $x_1-t\in R$.

Normalization dictionary with the published bimodule. The published The Soergel bimodule $B_i$ of a simple reflection has $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}=R^{\mathrm{\prime }}{\otimes }_{(R^{\mathrm{\prime }}{)}^{s_i}}{R}^{\mathrm{\prime }}(1)=B_i^{\mathrm{\prime }}\{-1\}$ because the Elias-Williamson shift $(1)$ is the internal shift $\{-1\}$ in the library convention, that is $M(r)=M\{-r\}$; equivalently $B_i^{\mathrm{\prime }}=B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}(-1)=B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}\{1\}\mathrm{.}$ The dictionary $M(r)=M\{-r\}$ is used throughout this page, and the generator $1\otimes 1$ of $B_i^{\mathrm{l}\mathrm{i}\mathrm{b}}$ has degree $-1$ while the generator $1\otimes 1$ of $B_i^{\mathrm{\prime }}$ has degree $0$.

Source convention. Khovanov writes the polynomial factor as $\mathbb{Q}[x_1]$. The ring splitting $R^{\mathrm{\prime }}=R[x_1]$ is valid, but a literal trivial factor for the bimodule actions uses a common invariant coordinate $t$ as above. The change of coordinates $x_1=t+(x_1-t)$ translates the source's ring notation into this bimodule description.

No choice principle is used in this definition.

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.

Let $F(\sigma )$ be the generator complex of Khovanov's generator complexes for the HHH construction, a bounded complex of graded $(R,R)$-bimodules with degree-zero differentials, and let $H{H}_h(R,-)$ be Hochschild homology, computed by the chain complexes $C_{\bullet }(R,-)$ of Hochschild chains and Hochschild homology with coefficients. Since every differential $d^j\text{\hspace{0.17em}⁣}:F(\sigma {)}^j\to F(\sigma {)}^{j+1}$ is a map of $R$-bimodules, it commutes with the Hochschild faces and induces a chain map $C_{\bullet }(R,F(\sigma {)}^j)\to C_{\bullet }(R,F(\sigma {)}^{j+1})$; since $d^{j+1}{d}^j=0$, the induced maps give a cochain complex of graded $\mathbb{Q}$-vector spaces for every $h\ge 0$, $(H{H}_h(R,F(\sigma {)}^{\bullet }),H{H}_h(R,d^{\bullet })),$ the termwise Hochschild complex in Hochschild degree $h$ (Termwise Hochschild homology and iterated homology applied to the bounded complex of $R$-bimodules $F(\sigma )$; its internal grading is inherited). The HHH groups of $\sigma$ are $HH{H}^{j,h,d}(\sigma ):=H^j(H{H}_h(R,F(\sigma {)}^{\bullet }){)}_d,j\in \mathbb{Z},h\ge 0,d\in \mathbb{Z},$ the cohomology in Rouquier degree $j$ and internal degree $d$ of the termwise complex, a trigraded $\mathbb{Q}$-vector space.

This is the componentwise construction of Beliakova-Putyra-Wehrli, section 3.8.6, equation (3.44) (Khovanov's construction is the case of the Rouquier complex of a braid word), and it is not the total Hochschild hyperhomology of the complex $F(\sigma )$ of Hochschild hyperhomology of a bounded bimodule complex: the latter is the abutment of a spectral sequence whose second page consists of the iterated homology groups of the termwise complex (Termwise Hochschild spectral sequence of a bounded bimodule complex), so the termwise groups carry the finer $(j,h,d)$ trigrading while the hyperhomology retains only $j-h$ up to filtration.

Caveats. The definition uses the chain-level Hochschild complexes and is choice-free: the identification of $H{H}_h$ with ${\mathrm{Tor}}_h^{R^e}(R,-)$ assumes the Axiom of Choice and is not needed here. The trigrading $(j,h,d)$ (Rouquier degree, Hochschild degree, internal degree) is not the trigrading $(a,q,t)$ of the comparison, which is fixed in The Koszul-Hochschild comparison respects crossing differentials and trigradings; the source's HHH is the cohomology of the termwise complex, that is the second page of the spectral sequence, not the hyperhomology abutment.

The difference between the termwise theory and the total Hochschild hyperhomology is worked out on the later examples page in Termwise and total Hochschild theories have different grading outputs ↗; nothing in the definition depends on that item.

Let $D$ be a marked braid diagram with nonempty closure, so its label count satisfies $m\ge 1$ and let $H(D)={⨁}_{j,k,l}{H}_{k,l}^j(D)$ be the triply graded groups of The Khovanov-Rozansky complex and trigraded braid homology, built over the polynomial ring $\mathbb{Q}[a,x_1,\dots ,x_m]$ generated by the coefficient variable $a$ and the $m$ strand labels. The reduced Khovanov-Rozansky homology $\bar{H}(D)$ is defined by repeating that construction verbatim with the label ring replaced by the ring of differences from one chosen coordinate, $\mathbb{Q}[a,x_1,\dots ,x_m]⇝\mathbb{Q}[a,x_2-x_1,\dots ,x_m-x_1],$ the coefficient $a$ being retained: the same marked MOY resolutions, arc and wide-edge factorizations, positive and negative crossing cones, removal of contractible summands and induced differentials, but with the $m$ strand labels replaced by the $m-1$ differences $x_2-x_1,\dots ,x_m-x_1$; the trivial variable of the one-mark circle is thereby dropped. The result $\bar{H}(D)={⨁}_{j,k,l}{\bar{H}}_{k,l}^j(D)$ is again a triply graded $\mathbb{Q}$-vector space, with the trigrading of that item.

By Khovanov-Rozansky II, end of section 1, one has a trigraded isomorphism $H(D)\cong \bar{H}(D){\otimes }_{\mathbb{Q}}\mathbb{Q}[x]$ in which $x$ is the trivial variable, and in the reduced theory the unknot has one-dimensional homology. The splitting is a statement about the two constructions, not a definition of $\bar{H}(D)$.

Grading normalization and the unknot. In the trigrading of The Khovanov-Rozansky complex and trigraded braid homology the one-mark circle has cohomology $\mathbb{Q}[x]\{-1,1\}$ at the resolution level (the companion computation of the matrix-factorization page, indexed there by the first and second bigrading); the reduction removes the trivial factor, leaving the one-dimensional reduced group in bidegree $(-1,1)$ at that level, and in the full trigraded theory the unknot class of the reduced homology sits in the raw tridegree $(-1,1,0)$ in $(k,l,j)$ order, whose image under the global correction $(1,-1,0)$ of The Koszul-Hochschild comparison respects crossing differentials and trigradings is the class $(0,0,0)$.

Caveats. $\bar{H}(D)$ is a construct of the reduced label ring and is not defined here by quotienting an arbitrary presentation of $H(D)$; the splitting $H(D)\cong \bar{H}(D){\otimes }_{\mathbb{Q}}\mathbb{Q}[x]$ is part of the source's assertion, recorded here with its grading conventions. The trivial variable $x$ is a polynomial variable carried by the one-mark circle and is not the coefficient variable $a$. Under AC (The Axiom of Choice), invariance of $\bar{H}(D)$ up to an overall trigrading shift is inherited from Khovanov-Rozansky braid homology is an oriented link invariant up to shift and is used in HHH is an oriented-link invariant up to an overall trigrading shift; the trigrading normalization is the one of The Khovanov-Rozansky complex and trigraded braid homology, and no Wu regrading is asserted here.

The coordinate reduction and polynomial splitting are choice-free. The oriented-link-invariance claim uses AC through the indicated Markov supplier and for homogeneous basis choices in step 3.1. The zero-strand tensor unit $\mathbb{Q}[a]$ has no chosen label $x_1$ and is excluded from this reduced construction.