ScalingStacks

Pick an oriented link $L$ and a component $L^{\prime}$ of $L.$ Let $L_{0}$ be $L$ with the orientation of $L^{\prime}$ reversed and let $l$ be the linking number of $L^{\prime}$ and $L\setminus L^{\prime}.$ Fixing a plane diagram $D$ of $L$, we count $l$ as half the number of double intersection points in $D$ of $L^{\prime}$ with $L\setminus L^{\prime}$ with weights $+1$ or $-1$ according to the following convention

Denote by $D_{0}$ the diagram $D$ with the reversed orientation of $L^{\prime}.$ Since $D_{0}$ and $D$ are the same as unoriented diagrams, $\overline{C}(D_{0})=\overline{C}(D).$ Also

We obtain

Let $K,K_{1}$ be oriented knots and $(-K)$ be $K$ with orientation reversed. In a similar fashion we deduce

Let $D$ be a diagram of an oriented link $L$ and denote by ${\mathrm{cm}}(L)$ the number of connected components of $L.$ Then it is easy to see that $C^{i}_{j}(D)=0$ if parities of $j$ and ${\mathrm{cm}}(L)$ differ. This observation implies