ScalingStacks

Proposition 34 For oriented knots $K_{1},K_{2},$ the isomorphism classes of the abelian groups ${\cal H}^{i,j}(K_{1}\sqcup K_{2}),{\cal H}^{i,j}(K_{1}\#K_{2})$ can be arranged into long exact sequences

$$\begin{array}[]{ccccccc}\to&{\cal H}^{i-1,j-1}(K_{1}\sqcup K_{2})&\to&{\cal H}^{i-1,j-2}(K_{1}\#K_{2})&\to&{\cal H}^{i,j}(K_{1}\#K_{2})&\to\\ \to&{\cal H}^{i,j-1}(K_{1}\sqcup K_{2})&\to&{\cal H}^{i,j-2}(K_{1}\#K_{2})&\to&{\cal H}^{i+1,j}(K_{1}\#K_{2})&\to\end{array}$$

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