ScalingStacks

A plane diagram $D$ of an oriented $(1,1)$-tangle $L$ is a generic projection of $L$ onto $\mathbb{R}\times[0,1].$

If $D$ is a plane diagram of an oriented $(1,1)$-tangle, define $x(D)$ and $y(D)$ in the same way as for plane diagrams of oriented links (see Section 2.4).

Fix a graded $A$-module $M.$ Let $n$ be the number of double points of $D$, so that $n=x(D)+y(D)$ and $\mathcal{I}$ the set of double points of $D.$ To $M$ and $D$ associate a commutative $\mathcal{I}$-cube $V_{D}^{M}$ over the category $A\mbox{-mod}_{0}$ of graded $A$-modules as follows.

For $\mathcal{L}\subset\mathcal{I}$ the $\mathcal{L}$-resolution $D(\mathcal{L})$ of $D$ consists of a disjoint union of circles and an interval. The functor $F_{M}$ (see Section 8.2) assigns a graded $A$-module to $D(\mathcal{L}).$ Define

Maps between $V_{D}^{M}(\mathcal{L})$ for various subsets $\mathcal{L}$ are defined by the procedure completely analogous to the one described in Section 4.2. Due to shifts $\{-|\mathcal{L}|\},$ these maps of graded $A$-modules are grading-preserving, rather than just graded maps, so that $V_{D}^{M}$ is a commutative cube over $A\mbox{-mod}_{0}.$

Example: For a diagram $D,$ depicted below,

resolutions of $D$ are

so that

and the structure map $V_{D}^{M}(\emptyset)\longrightarrow V_{D}^{M}(\mathcal{I})$ is the multiplication map $m_{M}:A\otimes M\to M\{-1\}.$

Next we transform the commutative $\mathcal{I}$-cube $V_{D}^{M}$ into a skew commutative $\mathcal{I}$-cube by putting minus signs in front of some structure maps of $V_{D}^{M},$ or, equivalently, by tensoring it with $E_{\mathcal{I}}.$ Denote by $\overline{C}_{M}(D)$ the complex $\overline{C}(V_{D}^{M}\otimes E_{\mathcal{I}})$ of graded $A$-modules. Define

Denote the $i$-th cohomology group of the complex $C_{M}(D)$ by $H^{i}(D,M).$ These cohomology groups are graded $A$-modules. Denote the $j$-th graded component of $H^{i}(D,M)$ by $H^{i,j}(D,M)$

Our proof of Theorem 1 immediately generalizes without essential modifications to a proof of Theorem 3. Proposition 38 is used to establish direct sum decompositions of $C_{M}(D)$, analogous to decompositions of $C(D)$, for suitable $D,$ given by Propositions 11, 14, 18.1, 21.1.

$\square$

Cohomology groups $H^{i}(D),$ defined in Section 4.2, is a special case of groups $H^{i}(D,M),$ as the next proposition explains.

$\square$

Given a finitely-generated graded $A$-module $M,$ define the graded Euler characteristic $\widehat{\chi}(M)$ by

$\square$

Given two graded $A$-modules $M,N$ and a grading-preserving homomorphism $f:M\to N$, it induces a map of commutative cubes $V_{D}^{M}\to V_{D}^{N},$ which, in turn, induces a map of complexes $C_{M}(D)\to C_{N}(D)$ and a map of cohomology groups $H^{i}(D,M)\to H^{i}(D,N).$ So, in fact, each diagram $D$ of an oriented long link defines functors $H^{i}_{D}$ from the category $A\mbox{-mod}_{0}$ of graded $A$-modules to itself, $H^{i}_{D}(M)=H^{i}(D,M).$ If two diagrams $D_{1},D_{2}$ are related by a Reidemeister move, constructions of Section 5 extend to functor isomorphism $H^{i}_{D_{1}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{i}_{D_{2}}.$ Let us frame this observation into a proposition.

Oriented long links with one component correspond one-to-one to oriented knots in $\mathbb{R}^{3}.$ Thus, Proposition 41 gives invariants of oriented knots in $\mathbb{R}^{3}.$ Moreover, if a diagram $D$ represents an oriented knot and $D^{\prime}$ is the diagram obtained from $D$ by reversing the orientation of the underlying curve, there is a natural in $M$ isomorphism $H^{i}(D,M)=H^{i}(D^{\prime},M).$ Consequently, for knots, isomorphism classes of functors $H^{i}_{D}$ do not depend on the orientation, and $H^{i}_{D}$ provide “functor-valued” invariants of non-oriented knots. Of course, these invariants depend on how the ambient 3-space is oriented.

Let $D$ be the 3-crossing diagram of the left-hand trefoil (knot $T_{2,3}$ in the notations of Section 6.2). The functors $H^{i}_{D}$ are written below

where ${\mathrm{ker}}\hskip 3.61371pt2X(M)=\{t\in M|2Xt=0\}.$