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1. Decomposing sutured monopole and Instanton Floer homologies

   Ghosh, Sudipta; Li, Zhenkun (November 2019, ArXivorg)

   In this paper generalizes the work of the second author 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 handle-bodies. 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 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. The same result was originally proved by Ozsváth and Szabó for link Floer homology. 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, 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 S3. This result was originally achieved by Kronheimer and Mrowka. 

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2. Direct systems and knot Floer homology

   Li, Zhenhun (January 2019, ArXiv.org)

   In this paper we construct possible candidates for the minus version of monopole or instanton knot Floer homology. We use a direct system which was introduced by Etnyre, Vela-Vick and Zarev. If K is a knot then we can construct a direct system based on a sequence of sutures on the boundary of the knot complement and the direct limit is of our interests. We prove that a Seifert surface of the knot will induce an Alexander grading and there is a U map on the direct limit shifting the degree down by 1. We prove some basic properties and compute the case of unknots. We also use the techniques developed in this paper to compute the sutured monopole and instanton Floer homology of a solid torus with any valid sutures. 

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3. Floer homology and right-veering monodromy

   https://doi.org/10.1515/crelle-2024-0079  

   Baldwin, John A; Ni, Yi; Sivek, Steven (October 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering.In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda–Kazez–Matić.Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms.We describe applications of this work to Dehn surgeries and taut foliations. 

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4. Sutured instanton homology and Heegaard diagrams

   https://doi.org/10.1112/S0010437X23007303  

   Baldwin, John A; Li, Zhenkun; Ye, Fan (September 2023, Compositio Mathematica)

   Suppose$$\mathcal {H}$$is an admissible Heegaard diagram for a balanced sutured manifold$$(M,\gamma )$$. We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology$$\mathit {SHI}(M,\gamma )$$. It follows, in particular, that strong L-spaces are instanton L-spaces. 

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5. Naturality and Mapping Class Groups in Heegaard Floer Homology

   https://doi.org/10.1090/memo/1338  

   Juhász, András; Thurston, Dylan; Zemke, Ian (September 2021, Memoirs of the American Mathematical Society)

   We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds. 

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