The Khovanov-Rozansky complex and trigraded braid homology

The tensor product is a complex with the stated potential. The differential of $C(D)$ is the sum, with the Koszul signs of the totalization, of the differentials of the factors $C_p$ and $C_c$. Each $C_p$ is a two-term complex by [F1] and each $C_c$ is a single factorization by [F2], so the totalized differential satisfies ${\mathrm{\partial }}^2=0$ strictly, and $C(D)$ is an object of $K({\mathrm{h}\mathrm{m}\mathrm{f}}_w)$. The square of the internal differential of a tensor product accumulates the individual potentials, so that of $C(D)$ is $a{\sum }_p{ϵ}_p{x}_p$ over the boundary points: every internal label occurs in exactly two local factors with opposite signs and cancels, exactly as for a marked graph in [F2]. For a closed braid diagram there are no boundary points and the potential is $w=0$, so $C(D)$ is a genuine complex of bigraded $\mathbb{Q}[a]$-modules.

Termwise cohomology and trivial $a$-action. Every resolution of the nonempty closed braid is a nonempty closed marked graph with at least one linear row: a crossingless circle has a mark and hence an arc factor, while each resolved crossing contributes arc or wide-edge factors. The row reduction of [F3] turns its first linear row into $(a,0)$ and leaves all other linear and quadratic rows with first entry zero. The explicit polynomial splitting of that first row over $\mathbb{Q}$ reduces its cohomology to the specialization $a=0$ of the remaining Koszul complex, with the first row's parity and internal shift retained. Thus $a$ acts trivially on every $C{H}^j(D)$ and these are bigraded $\mathbb{Q}$-vector spaces. This reduction does not imply finite rank over $\mathbb{Q}[a]$; the one-mark circle already leaves the polynomial variable $x$. Each resolution nevertheless has finite-dimensional pieces in each bigrading, since it uses finitely many polynomial variables of positive degrees and finitely many shifted Koszul terms.

The empty boundary case. For the zero-strand diagram there are no crossings or arc factors. The empty tensor is the graph tensor unit $\mathbb{Q}[a]$ of [F2], in degree $0$ with zero differential. Thus $C{H}^0=H^0=\mathbb{Q}[a]$ and all other outer degrees vanish. Its homogeneous monomials $a^d$ have bigrading $(2d,0)$, so each fixed bidegree is finite and the raw Euler series is ${\sum }_{d\ge 0}{t}^{2d}=(1-t^2{)}^{-1}$. Multiplication by $a$ is nonzero, as asserted separately.

The induced differential and the trigraded cohomology. The differential $\mathrm{\partial }$ of $C(D)$ is a sum of morphisms ${\chi }_0,{\chi }_1$ of bidegree $(0,0)$ between factorizations, so it commutes with the internal differentials of the terms, hence maps cycles to cycles and boundaries to boundaries in each term and induces a map $\mathrm{\partial }\text{\hspace{0.17em}⁣}:C{H}^j(D)\to C{H}^{j+1}(D)$. Since ${\mathrm{\partial }}^2=0$ on $C(D)$ by step 1.1, the induced maps satisfy ${\mathrm{\partial }}^2=0$ on $CH(D)$, and since they preserve the bigrading, the cohomology $H(D)={⨁}_{j,k,l}{H}_{k,l}^j(D)$ is triply graded with $j$ the cohomological degree and $k,l$ the two bigrading degrees; the Euler characteristic $⟨D⟩={\sum }_{j,k,l}(-1{)}^j{t}^k{q}^l{\dim }_{\mathbb{Q}}{H}_{k,l}^j(D)$ is therefore defined. This is the integer-graded construction of Khovanov-Rozansky II, section 1; invariance is established by later items.