ScalingStacks

Remark 1.4. Although the invariant $\mathcal{S}_{0}^{N}(W;L,\mathbbm{k})$ for any four-manifold $W$ can be expressed purely in terms of link homology in $S^{3}$, specifically $\operatorname{KhR}_{N}$, it would be difficult to prove directly that these expressions yield a four-manifold invariant. A direct proof of invariance, without comparing to the intrinsically defined invariants $\mathcal{S}_{0}^{N}$, would require checking handle slide and handle cancellation moves as well as higher coherence conditions between their composites. Handle slides for 2-handles are studied (for $N=2$) in [15] and instances of $(2,3)$-handle cancellation are discussed in Example 3.8. Another interesting question concerns the behaviour of our algebraic decription of $\mathcal{S}_{0}^{N}(W;L,\mathbbm{k})$ under reversing the handle decomposition of $W$. However, our approach uses transversality arguments to isotope skeins away from cocores of handles to yield simplified handle formulas; hence, we do not expect these formulas to reflect the duality between $k$- and $(4-k)$-handles, because the duality does not respect cocores.