Decomposing sutured monopole and instanton Floer homologies

In this paper, we generalize the work of second author and prove a grading shifting property, in sutured monopole instanton Floer theories, for general balanced manifolds. This result has few consequences. First, offer an algorithm that computes homologies family handle-bodies. Second, obtain canonical decomposition build polytopes these two which was initially achieved by Juh\'asz (Heegaard) homology. Third, establish Thurston-norm detection knot homologies, were introduced Kronheimer Mrowka. The same originally proved Ozsv\'ath Szab\'o link Last, construction minus versions homology, done knots author, to case links. Along with polytopes, also that, manifold vanishing rank or homology bounds depth manifold. As corollary, independent proof as mentioned above, both detect fibred $S^3$.