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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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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.more » « less
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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.more » « less
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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.more » « less
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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.more » « less
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