ScalingStacks

Setting $c=0$ and taking $\mathbb{Z}$ instead of $R=\mathbb{Z}[c]$ as the base ring, everything we did in Sections 2, 4 and 5 goes through in exactly the same manner. The role of the ring $A$ will be played by the free graded abelian group $\cal A$ of rank $2$ with generators $\mathbf{1}$ and $X$ in degrees $1$ and $-1$ correspondingly. $\cal A$ has commutative algebra and cocommutative coalgebra structures:

and the identity (16) holds. By abuse of notations, we use $m$ and $\Delta$ to denote multiplication and comultiplication in ${\cal A}.$ Earlier $m$ and $\Delta$ were used to denote multiplication and comultiplication in $A.$ As in Section 2.3, we construct a functor $\cal F$ from the category $\mathcal{M}$ of closed one-manifolds and cobordisms between them to the category of graded abelian groups and graded homomorphisms. To a disjoint union of $k$ circles functor $\cal F$ assigns the group ${\cal A}^{\otimes k}.$ To elementary surfaces $S_{2}^{1},S_{1}^{2},S_{0}^{1},S_{1}^{0},S_{2}^{2}$ and $S_{1}^{1}$ (see Section 2.3) functor $\cal F$ assigns maps $m,\Delta,\iota,\epsilon,\mbox{Perm}$ and Id between suitable tensor powers of ${\cal A}.$ The maps $\iota:\mathbb{Z}\to{\cal A}$ and $\epsilon:{\cal A}\to\mathbb{Z}$ are given by

while Perm is just the permutation map ${\cal A}\otimes{\cal A}\to{\cal A}\otimes{\cal A}.$

To a diagram $D$ of an oriented link $L$ we can then associate a commutative $\mathcal{I}$-cube ${\cal V}_{D}$ of graded abelian groups and grading-preserving homomorphism, by the same procedure as the one described in Section 4.2, using the functor $\cal F$ instead of $F$. In particular, for $\mathcal{L}\subset\mathcal{I}$ we have ${\cal V}_{D}(\mathcal{L})={\cal F}(D(\mathcal{L}))\{-|\mathcal{L}|\}$ where $\{k\}$ shifts the grading down by $k.$

Let $\cal{AB}$ be the category of graded abelian groups and grading-preserving homomorphisms. Let ${\cal E}_{\mathcal{I}}$ be the skew-commutative $\mathcal{I}$-cube $E_{\mathcal{I}}\otimes_{R}\mathbb{Z}$ over $\cal{AB}.$

Tensoring ${\cal V}_{D}$ with ${\cal E}_{\mathcal{I}}$ over $\mathbb{Z},$ we get a skew commutative $\mathcal{I}$-cube ${\cal V}_{D}\otimes{\cal E}_{\mathcal{I}}$ over the category $\cal{AB}.$ From this skew-commutative $\mathcal{I}$-cube we get a complex $\overline{{\cal C}}({\cal V}_{D}\otimes{\cal E}_{\mathcal{I}})$ of graded abelian groups with a grading-preserving differential (Section 3.4). Denote this complex by $\overline{\cal C}(D)$ and by ${\cal C}(D)$ the shifted complex

If we consider $\mathbb{Z}$ as a graded $R$-module, concentrated in degree $0,$ so that $c\mathbb{Z}=0,$ then

To a plane diagram $D$ of an oriented link $L$ we thus associate a complex of graded abelian groups ${\cal C}(D)$. Denote the $i$-th cohomology group of the $j$-th graded summand of ${\cal C}(D)$ by ${\cal H}^{i,j}(D).$ These cohomology groups are finitely-generated abelian groups. For each diagram $D$ as we vary $i$ and $j$ over all integers, only a finite number of these groups are non-zero.

Proof: Set $c=0$ in the proof of Theorem 1. $\square$

For a diagram $D$ of the link $L,$ denote the isomorphism classes of ${\cal H}^{i,j}(D)$ by ${\cal H}^{i,j}(L).$

Denote by $\overline{{\cal C}}^{i}(D)$ (respectively, ${\cal C}^{i}(D)$) the $i$-th group of the complex $\overline{{\cal C}}(D)$ (respectively, ${\cal C}(D)$) and by $\overline{{\cal C}}^{i}_{j}(D)$ (respectively, by ${\cal C}^{i}_{j}(D)$) the $j$-th graded component of $\overline{{\cal C}}^{i}(D)$ (respectively, ${\cal C}^{i}(D)$), so that $\overline{{\cal C}}^{i}(D)={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}\overline{{\cal C}}^{i}_{j}(D),$ respectively, ${\cal C}^{i}(D)={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}{{\cal C}}^{i}_{j}(D).$ For an diagram $D$ denote by ${\cal H}^{i}(D)$ the graded abelian group ${\mathop{\oplus}\limits_{j\in\mathbb{Z}}}{\cal H}^{i,j}(D).$ In other words, ${\cal H}^{i}(D)$ is the $i$-th cohomology group of ${\cal C}(D).$ Denote by $\overline{{\cal H}}^{i}(D)$ the $i$-th cohomology group of the complex $\overline{\cal C}(D)$ and by $\overline{{\cal H}}^{i,j}(D)$ the $j$-th graded component of $\overline{{\cal H}}^{i}(D),$ so that $\overline{{\cal H}}^{i}(D)={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}\overline{{\cal H}}^{i,j}(D).$