Publications (68)28.88 Total impact
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ABSTRACT: We modify the construction of knot Floer homology to produce a oneparameter family of homologies for knots in the threesphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the fourball genus and the concordance genus of knots. We give some applications of these homomorphisms.07/2014;  [Show abstract] [Hide abstract]
ABSTRACT: Given a link in the threesphere, Ozsv\'ath and Szab\'o showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of bimodules associated to Dehn twists, and a general pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on computing the bimodules; this part focuses on the pairing theorem for polygons, in order to prove that the spectral sequence constructed in the previous part agrees with the one constructed by Ozsv\'ath and Szab\'o.04/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting.01/2013;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we show how to recover the relative Qgrading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.11/2012; 
Article: Notes on bordered Floer homology
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ABSTRACT: This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HFhat to 3manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.11/2012; 
Article: Knots in lattice homology
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ABSTRACT: Assume that \Gamma_{v_0} is a tree with vertex set Vert(\Gamma_{v_0})={v_0, v_1,..., v_n}, and with an integral framing (weight) attached to each vertex except v_0. Assume furthermore that the intersection matrix of G=\Gamma_{v_0}{v_0} is negative definite. We define a filtration on the chain complex computing the lattice homology of G and show how to use this information in computing lattice homology groups of a negative definite graph we get by attaching some framing to v_0. As a simple application we produce families of graphs which have arbitrarily many bad vertices for which the lattice homology groups are shown to be isomorphic to the corresponding Heegaard Floer homology groups.08/2012; 
Article: Knot lattice homology in Lspaces
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ABSTRACT: We show that the knot lattice homology of a knot in an Lspace is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that Gw is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)$ is isomorphic to the Heegaard Floer homology HF^(Y_G) of the corresponding rational homology sphere Y_G.07/2012;  [Show abstract] [Hide abstract]
ABSTRACT: Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heegaard Floer homology of the underlying 3manifold.06/2012; 
Article: Tour of bordered Floer theory.
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ABSTRACT: Heegaard Floer theory is a kind of topological quantum field theory (TQFT), assigning graded groups to closed, connected, oriented 3manifolds and group homomorphisms to smooth, oriented fourdimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3manifolds with boundary, with extendedTQFTtype gluing properties. In this survey, we explain the formal structure and construction of bordered Floer homology and sketch how it can be used to compute some aspects of Heegaard Floer theory.Proceedings of the National Academy of Sciences 05/2011; 108(20):808592. · 9.81 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin^cstructure is faithful. The paper is selfcontained, and in particular does not assume any background in Floer homology. Comment: 26 pages, 9 figures12/2010;  [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: Given a link in the threesphere, Z. Szab\'o and the second author constructed a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched doublecover. The aim of this paper and its sequel is to explicitly calculate this spectral sequence, using bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of filtered bimodules associated to Dehn twists and a pairing theorem for polygons. In this paper we give the first ingredient, and so obtain a combinatorial spectral sequence from Khovanov homology to Heegaard Floer homology; in the sequel we show that this spectral sequence agrees with the previously known one. Comment: 43 pages, 16 figures11/2010;  [Show abstract] [Hide abstract]
ABSTRACT: Bordered Heegaard Floer homology is an invariant for threemanifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed threemanifold, as well as the bordered Floer homology of any 3manifold with boundary.10/2010;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukayacategorical constructions. The result also leads to various dualities in bordered Floer homology.05/2010;  [Show abstract] [Hide abstract]
ABSTRACT: Bordered Heegaard Floer homology is a threemanifold invariant which associates to a surface F an algebra A(F) and to a threemanifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two version of the 3manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology.03/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We consider a stabilized version of hat Heegaard Floer homology of a 3manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance properties.12/2009;  [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: We show that every 3manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially. Comment: Fixed figures11/2008;  [Show abstract] [Hide abstract]
ABSTRACT: We construct Heegaard Floer theory for 3manifolds with connected boundary. The theory associates to an oriented, parametrized twomanifold a differential graded algebra. For a threemanifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an Ainfinity module. Both are welldefined up to chain homotopy equivalence. For a decomposition of a 3manifold into two pieces, the Ainfinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of threemanifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a threemanifold with torus boundary with the knot Floer homology of a filling.10/2008;  [Show abstract] [Hide abstract]
ABSTRACT: We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications. Comment: 25 pages, 9 figures10/2008;
Publication Stats
3k  Citations  
28.88  Total Impact Points  
Top Journals
Institutions

2003–2013

Princeton University
 Department of Mathematics
Princeton, New Jersey, United States


2011

Massachusetts Institute of Technology
 Department of Mathematics
Cambridge, MA, United States


2001–2011

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