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17
FUNCTORIALITY FOR LAGRANGIAN CORRESPONDENCES IN FLOER THEORY
"... We generalize Lagrangian Floer theory to sequences of Lagrangian correspondences and establish an isomorphism between the Floer homology groups of sequences that are related by the geometric composition of Lagrangian correspondences. On these Floer homologies, we define relative invariants arising ..."
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Cited by 49 (10 self)
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We generalize Lagrangian Floer theory to sequences of Lagrangian correspondences and establish an isomorphism between the Floer homology groups of sequences that are related by the geometric composition of Lagrangian correspondences. On these Floer homologies, we define relative invariants arising from “quilted pseudoholomorphic surfaces”: Collections of pseudoholomorphic maps to various target spaces with “seam conditions” in Lagrangian correspondences and boundary conditions in Lagrangian submanifolds. Using these new invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence we obtain “categorification commutes with composition”
Khovanov homology is an unknotdetector
"... Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then s ..."
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Cited by 22 (2 self)
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Abstract. We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot. 1
A colored sl(n)homology for links in S³
, 2009
"... Fix an integer N ≥ 2. To each diagram of a link colored by 1,..., N, we associate a chain complex of graded matrix factorizations. We prove that the homopoty type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is is ..."
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Cited by 19 (6 self)
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Fix an integer N ≥ 2. To each diagram of a link colored by 1,..., N, we associate a chain complex of graded matrix factorizations. We prove that the homopoty type of this chain complex is invariant under Reidemeister moves. When every component of the link is colored by 1, this chain complex is isomorphic to the chain complex defined by Khovanov and Rozansky in [17]. We call the homology of this chain complex the colored sl(N)homology and conjecture that it decategorifies to the quantum sl(N)polynomial
QUILTED FLOER COHOMOLOGY
"... Abstract. We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic 2 ..."
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Cited by 18 (4 self)
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Abstract. We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic 2category as well as for the definition of topological invariants via decomposition and representation in the symplectic category. Here we give some first direct symplectic applications: Calculations of Floer cohomology, displaceability of Lagrangian correspondences, and transfer of displaceability under
Knots, sutures and excision
"... Abstract. We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots. Contents ..."
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Cited by 13 (0 self)
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Abstract. We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots. Contents
Instanton Floer homology and the Alexander polynomial
 Algebr. Geom. Topol
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 9 (2 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Supersymmetric Surface Operators, FourManifold Theory And Invariants
 In Various Dimensions, Adv.Theor.Math.Phys. 15 (2011) 071–130, [arXiv:1006.3313
"... We continue our program initiated in [1] to consider supersymmetric surface operators in a topologicallytwisted N = 2 pure SU(2) gauge theory, and apply them to the study of fourmanifolds and related invariants. Elegant physical proofs of various seminal theorems in fourmanifold theory obtained b ..."
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Cited by 5 (0 self)
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We continue our program initiated in [1] to consider supersymmetric surface operators in a topologicallytwisted N = 2 pure SU(2) gauge theory, and apply them to the study of fourmanifolds and related invariants. Elegant physical proofs of various seminal theorems in fourmanifold theory obtained by OzsváthSzabo ́ [2, 3] and Taubes [4], will be furnished. In particular, we will show that Taubes ’ groundbreaking and difficult result – that the ordinary SeibergWitten invariants are in fact the Gromov invariants which count pseudoholomorphic curves embedded in a symplectic fourmanifold X – nonetheless lends itself to a simple and concrete physical derivation in the presence of “ordinary ” surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and “ramified ” SeibergWitten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its threesubmanifold. Via these identities, and a physical formulation of the “ramified ” Donaldson invariants of fourmanifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from “ramified ” instantons by KronheimerMrowka [5]; and (ii) monopole Floer homology and SeibergWitten theory on symplectic fourmanifolds by KutluhanTaubes [4, 6]. Supersymmetry, as well as other physical concepts such as Rinvariance, electricmagnetic duality, spontaneous gauge symmetrybreaking and localization onto supersymmetric configurations in topologicallytwisted quantum field theories, play a pivotal role in our story. ∗email:
Gauge theory and Rasmussen’s invariant
"... For a knot K S 3, the (smooth) slicegenus g.K / is the smallest genus of any properly embedded, smooth, oriented surface † B4 with boundary K. In [12], Rasmussen used a construction based on Khovanov homology to define a knotinvariant s.K / 2 2Z with the following properties: ..."
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Cited by 4 (0 self)
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For a knot K S 3, the (smooth) slicegenus g.K / is the smallest genus of any properly embedded, smooth, oriented surface † B4 with boundary K. In [12], Rasmussen used a construction based on Khovanov homology to define a knotinvariant s.K / 2 2Z with the following properties:
Floer field theory for tangles
"... Abstract. We construct functorvalued invariants invariants for cobordisms possibly containing tangles and certain trivalent graphs. The latter gives gaugetheoretic invariants similar to KhovanovRozansky homology [12]. Contents ..."
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Cited by 4 (4 self)
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Abstract. We construct functorvalued invariants invariants for cobordisms possibly containing tangles and certain trivalent graphs. The latter gives gaugetheoretic invariants similar to KhovanovRozansky homology [12]. Contents
The pillowcase and perturbations of traceless representations of knot groups
 Geom. Topol
"... Abstract. We introduce explicit holonomy perturbations of the ChernSimons functional on a 3ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka’s singular ins ..."
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Abstract. We introduce explicit holonomy perturbations of the ChernSimons functional on a 3ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka’s singular instanton knot homology nondegenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the fourpunctured 2sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand nontrivial differentials in the spectral sequence from Khovanov homology to singular instanton