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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. Instantons and Bar-Natan homology

   https://doi.org/10.1112/s0010437x2000768x  

   Kronheimer, P. B.; Mrowka, T. S. (March 2021, Compositio Mathematica)

   A spectral sequence is established whose $$E_{2}$$ page is Bar-Natan's variant of Khovanov homology and which abuts to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence whose $$E_{2}$$ page is a characteristic-2 version of $$F_{5}$$ homology in Khovanov's classification. 

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4. 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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5. SU(3) instanton homology for webs and foams

   https://doi.org/10.48550/arxiv.2505.02755  

   Kronheimer, Peter B; Mrowka, Tomasz S (May 2025, AxRiV)

   An instanton homology is constructed for webs and foams, using gauge theory with structure group SU (3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU (3) homology counts Tait colorings when theweb is planar. Unlike the SO(3) case, the SU (3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed. 

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