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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. Instanton and the depth of taut foliations

   Li, Zhenkun (July 2020, ArXivorg)

   Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured manifold. The same argument can be adapted to the monopole and even the Heegaard Floer settings, which gives a partial answer to one of Juhasz's conjectures. Using the nature of instanton Floer homology, on knot complements, we can construct a taut foliation with bounded depth, given some information on the representation varieties of the knot fundamental groups. This indicates a mystery relation between the representation varieties and some small depth taut foliations on knot complements, and gives a partial answer to one of Kronheimer and Mrowka's conjecture. 

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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. On the tau invariants in instanton and monopole Floer theories

   https://doi.org/10.1112/topo.12346  

   Ghosh, Sudipta; Li, Zhenkun; Wong, C‐M Michael (June 2024, Journal of Topology)

   Abstract We unify two existing approaches to thetauinvariants in instanton and monopole Floer theories, by identifying , defined by the second author via theminusflavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3‐manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots. 

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5. 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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