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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.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 is the second paper of this series. We define the monopole Floer homol- ogy for 3-manifolds with torus boundary, extending the work of Kronheimer-Mrowka for closed 3-manifolds. The Euler characteristic of this Floer homology recovers the Milnor torsion invariant of the 3-manifold by a theorem of Meng-Taubes.more » « less
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The end point of this series of papers is to construct the monopole Floer ho- mology for 3-manifolds with torus boundary. In the first paper, we explain the idea from the standpoint of gauged Landau-Ginzburg models and address a few model problems related to the compactness of moduli spaces, using a Bochner-type formula associated to the gauged Witten equations.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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