Subsection 7.6 (0PNN) · ScalingStacks

7.6 Examples

Perhaps the graded groups ${\cal H}^{i}(D)$ are easier to compute than $H^{i}(D).$ The latter are computed via the complex $C(D)$ of free graded $R$ -modules, and a complex for ${\cal H}^{i}(D)$ is obtained by tensoring $C(D)$ with $\mathbb{Z}$ over $R,$ so that free graded $R$ -modules become free abelian groups of the same rank. In practice, the computation of ${\cal H}^{i}(D)$ faces the problem of effectively simplifying complexes of abelian groups of exponentially high rank. The shortcut for the computation of $H^{i}(D)$ described in Section 6.2 works equally well for groups ${\cal H}^{i}(D),$ with Proposition 25 generalized to complexes $\overline{{\cal C}}(D).$ This easily leads to a computation of cohomology groups ${\cal H}^{i,j}(T_{2,k})$ of the $(2,k)$ torus link $T_{2,k},$ oriented as in Section 6.2.

0PMD

Proposition 35 Cohomology groups ${\cal H}^{i,j}(T_{2,k}),k>1$ are isomorphic to

\begin{array}[]{lll}{\cal H}^{0,-k}(T_{2,k})&=&\mathbb{Z},\\ {\cal H}^{0,2-k}(T_{2,k})&=&\mathbb{Z},\\ {\cal H}^{-2j-1,-4j-2-k}(T_{2,k})&=&\mathbb{Z}\hskip 14.45377pt\mbox{ for }1\leq j\leq\frac{k-1}{2},j\in\mathbb{Z},\\ {\cal H}^{-2j,-4j-k}(T_{2,k})&=&\mathbb{Z}_{2}\hskip 14.45377pt\mbox{ for }1\leq j\leq\frac{k-1}{2},j\in\mathbb{Z},\\ {\cal H}^{-2j,-4j+2-k}(T_{2,k})&=&\mathbb{Z}\hskip 14.45377pt\mbox{ for }1\leq j\leq\frac{k-1}{2},j\in\mathbb{Z},\\ {\cal H}^{-k,-3k}(T_{2,k})&=&\mathbb{Z}\hskip 14.45377pt\mbox{ for even }k,\\ {\cal H}^{-k,2-3k}(T_{2,k})&=&\mathbb{Z}\hskip 14.45377pt\mbox{ for even }k,\\ {\cal H}^{i,j}(T_{2,k})&=&0\hskip 14.45377pt\mbox{ for all other values of }i\mbox{ and }j\end{array}

Mikhail Khovanov

Original source: arXiv:math/9908171v2