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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. Tau invariants in monopole and instanton theories

   Li, Zhenkun (February 2020, ArXivorg)

   This paper introduces tau invariants coming from the minus versions of monopole and instanton theory for knots in S3 recently defined by Li. Some basic properties are proved such as concordant invariance. The paper computes the minus versions of monopole and instanton knot Floer homologies for twist knots. 

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3. Knot Floer homologies in monopole and instanton theory via sutures.

   Li, Zhenkun (November 2019, ArXivorg)

   This paper constructs possible candidates for the minus versions of monopole and instanton knot Floer homologies. For a null-homologous knot K insid Y and a base point p on K, we can associate the minus versions, KHM^{-}(Y, K, p) and KHI^{-}(Y, K, p), to the triple (Y, K, p). We prove that a Seifert surface of K induces a Z-grading, and there is an U-map on the minus versions, which is of degree -1. We also prove other basic properties of them. If K inside Y is not null-homologous but represents a torsion class, then we can also construct the corresponding minus versions for (Y,K,p). We also proved a surgery-type formula relating the minus versions of a knot K with those of the dual knot, when performing a Dehn surgery of large enough slope along K. The techniques developed in this paper can also be applied to compute the sutured monopole and instanton Floer homologies of any sutured solid tori. 

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4. 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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5. Stabilization distance bounds from link Floer homology

   https://doi.org/10.1112/topo.12338  

   Juhász, András; Zemke, Ian (June 2024, Journal of Topology)

   Abstract We consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks. 

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