Publications (21)5.96 Total impact
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ABSTRACT: This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three and fourdimensional topology via surgery formulas. We also describe some conjectural relations to KhovanovRozansky homology.01/2014; 
Article: Cornered Heegaard Floer homology
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ABSTRACT: Bordered Floer homology assigns invariants to 3manifolds with boundary, such that the Heegaard Floer homology of a closed 3manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3manifolds with codimension2 corners, and prove that the bordered Floer homology of a 3manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.08/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We prove Furutatype bounds for the intersection forms of spin cobordisms between homology 3spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)equivariant SeibergWitten Floer Ktheory. In the process we introduce the notion of a Floer K_Gsplit homology sphere; this concept may be useful in an approach to the 11/8 conjecture.05/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We define Pin(2)equivariant SeibergWitten Floer homology for rational homology 3spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integervalued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4manifold. By previous work of GalewskiStern and Matumoto, this implies the existence of nontriangulable highdimensional manifolds.03/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3manifolds and 4manifolds are algorithmically computable (mod 2).10/2012;  [Show abstract] [Hide abstract]
ABSTRACT: Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its E_1 page is isomorphic to the HOMFLYPT chain complex of Khovanov and Rozansky. At the level of each E_1 summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.07/2011;  [Show abstract] [Hide abstract]
ABSTRACT: Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2algebra, the nilCoxeter sequential 2algebra, and to a surface with connected boundary an algebramodule over this 2algebra, such that a natural gluing property is satisfied. Moreover, with a view toward the structure of a potential Floer homology theory of 3manifolds with codimensiontwo corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.04/2011;  [Show abstract] [Hide abstract]
ABSTRACT: Let L be a link in an integral homology threesphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) the link Floer homology of L and all its sublinks, together with several chain maps between these complexes. Further, we introduce a way of presenting closed fourmanifolds with b_2^+ > 1 by fourcolored framed links in the threesphere. Given a link presentation of this kind for a fourmanifold X, we then describe the OzsvathSzabo mixed invariants of X in terms of a complete system of hyperboxes for the link. Finally, we explain how a grid diagram produces a particular complete system of hyperboxes for the corresponding link.11/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary threemanifolds (with coefficients in Z/2). The descriptions are based on presenting the threemanifold as an integer surgery on a link in the threesphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 OzsvathSzabo mixed invariants of closed fourmanifolds, in terms of grid diagrams.10/2009;  [Show abstract] [Hide abstract]
ABSTRACT: Given a Heegaard splitting of a 3manifold, we use Lagrangian Floer homology to construct a relatively Z/8Zgraded abelian group, which we conjecture to be a 3manifold invariant. Our motivation is to have a welldefined symplectic side of the AtiyahFloer Conjecture, for arbitrary 3manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the twoform on the resulting manifold has degeneracies, but we show that one can still develop a version of Floer homology in this setting. Comment: minor changes; final version11/2008;  [Show abstract] [Hide abstract]
ABSTRACT: Quasialternating links are a natural generalization of alternating links. In this paper, we show that quasialternating links are "homologically thin" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing.09/2007;  [Show abstract] [Hide abstract]
ABSTRACT: For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doublygraded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.Advances in Mathematics 05/2007; · 1.37 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for threemanifolds. Given a cobordism between two connected threemanifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.Duke Mathematical Journal 12/2006; · 1.70 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Given a crossing in a planar diagram of a link in the threesphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasialternating knot, the total rank of its knot Floer homology is equal to the determinant of the knot.10/2006; 
Article: On combinatorial link Floer homology
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ABSTRACT: Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a selfcontained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients. Comment: Updated to final published version.10/2006;  [Show abstract] [Hide abstract]
ABSTRACT: Given a grid presentation of a knot (or link) K in the threesphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.08/2006;  [Show abstract] [Hide abstract]
ABSTRACT: Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3sphere branched over a knot K, we obtain an invariant delta of knot concordance. We show that delta is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all Hthin knots. We also use delta to prove that for all knots K with tau(K)>0, the positive untwisted double of K is not smoothly slice.09/2005;  [Show abstract] [Hide abstract]
ABSTRACT: Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra $sl_{2m}.$ We exhibit bijections between a set of generators for the SeidelSmith cochain complex, the generators in Bigelow's picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.Duke Mathematical Journal 12/2004; · 1.70 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In a previous paper we have constructed an invariant of fourdimensional manifolds with boundary in the form of an element in the stable homotopy group of the SeibergWitten Floer spectrum of the boundary. Here we prove that when one glues two fourmanifolds along their boundaries, the BauerFuruta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.Journal of differential geometry 12/2003; · 1.18 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Using Furuta's idea of finite dimensional approximation in SeibergWitten theory, we refine SeibergWitten Floer homology to obtain an invariant of homology 3spheres which lives in the S^1equivariant graded suspension category. In particular, this gives a construction of SeibergWitten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of fourmanifolds with boundary which generalizes the BauerFuruta stable homotopy invariant of closed fourmanifolds.Geometry & Topology  GEOM TOPOL. 01/2003; 7:889932.
Publication Stats
364  Citations  
5.96  Total Impact Points  
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2007

Columbia University
 Department of Mathematics
New York City, NY, United States
