Abstract
In this paper, we generalize the work of the second author in Li (Direct systems and the knot monopole Floer homology, 2019. arXiv:1901.06679) and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handlebodies.. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász (Geom Topol 14(3):1303–1354, 2010) for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka (J Differ Geom 84(2):301–364, 2010). The same result was originally proved by Ozsváth and Szabó for link Floer homology in Ozsváth and Szabó (J Am Math Soc 21(3):671–709, 2008). Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author in Li (2019), to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in \(S^3\). This result was originally achieved by Kronheimer and Mrowka (2010).
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Acknowledgements
The authors would like to thank their advisors David Shea Vela-Vick and Tomasz Mrowka, for their enormous help. The authors would like to thank András Juhász and Yi Ni for helpful comments or conversations. The first author was supported by his advisor David Shea Vela-Vick’s NSF Grant 1907654 and Simons Foundation Grant 524876. The second author was supported by his advisor Tom Mrowka’s NSF Grant 1808794.
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