ScalingStacks

Proposition 40 Let $M$ be a finitely-generated graded $A$-module, $L$ an oriented $(1,1)$-tangle and $D$ a diagram of $L$. Then

$$\frac{K({\mathrm{cl}}(L))\widehat{\chi}(M)}{q+q^{-1}}=\sum_{i,j\in\mathbb{Z}}(-1)^{i}q^{j}{\mathrm{dim}}_{\mathbb{Q}}(H^{i,j}(D,M)\otimes_{\mathbb{Z}}\mathbb{Q}).$$

(207)

that is, the Kauffman bracket of ${\mathrm{cl}}(L)$ is proportional to the Euler characteristic of groups $H^{i,j}(D,M).$