Publications (37)0 Total impact
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ABSTRACT: Let L be a link in an integral homology three-sphere. 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 four-manifolds with b_2^+ > 1 by four-colored framed links in the
three-sphere. Given a link presentation of this kind for a four-manifold X, we
then describe the Ozsvath-Szabo 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;
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ABSTRACT: We give combinatorial descriptions of the Heegaard Floer homology groups for
arbitrary three-manifolds (with coefficients in Z/2). The descriptions are
based on presenting the three-manifold as an integer surgery on a link in the
three-sphere, and then using a grid diagram for the link. We also give
combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of
closed four-manifolds, in terms of grid diagrams.
10/2009;
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ABSTRACT: We show that every 3--manifold 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 figures
11/2008;
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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 3-manifolds 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 figures
10/2008;
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ABSTRACT: We construct Heegaard Floer theory for 3-manifolds with connected boundary.
The theory associates to an oriented, parametrized two-manifold a differential
graded algebra. For a three-manifold 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 A-infinity module. Both are
well-defined up to chain homotopy equivalence. For a decomposition of a
3-manifold into two pieces, the A-infinity 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
three-manifolds 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 three-manifold with torus boundary with the knot Floer homology
of a filling.
10/2008;
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ABSTRACT: We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3. Comment: Corrected naturality discussion.
03/2008;
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ABSTRACT: We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.
11/2007;
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ABSTRACT: Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating 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;
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ABSTRACT: The aim of this paper is to study the skein exact sequence for knot Floer homology. We prove precise graded version of this sequence, and also one using $\HFm$. Moreover, a complete argument is also given purely within the realm of grid diagrams.
08/2007;
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ABSTRACT: We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.
06/2007;
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ABSTRACT: We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots. Comment: Minor revisions
05/2007;
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ABSTRACT: We exhibit pairs of transverse knots with the same self-linking number that are not transversely isotopic, using the recently defined knot Floer homology invariant for transverse knots and some algebraic refinements of it. Comment: Added explanations, corrected statement of Proposition 2
03/2007;
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ABSTRACT: Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots. Comment: 27 pages, 13 figures; v2: Expand and correct discussion of links
11/2006;
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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 self-contained 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;
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ABSTRACT: Given a grid presentation of a knot (or link) K in the three-sphere, 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;
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ABSTRACT: We give a Dehn surgery characterization of the trefoil and the figure eight knots. These results are gotten by combining surgery formulas in Heegaard Floer homology from an earlier paper with the characterization of these knots in terms of their knot Floer homology given in a recent paper of Ghiggini.
05/2006;
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ABSTRACT: We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emphasize the application to the Thurston norm, and illustrate the theory in the case of the Conway link.
03/2006;
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ABSTRACT: We show that link Floer homology detects the Thurston norm of a link complement. As an application, we show that the Thurston polytope of an alternating link is dual to the Newton polytope of its multi-variable Alexander polynomial. To illustrate these techniques, we also compute the Thurston polytopes of several specific link complements.
02/2006;
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01/2006;
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ABSTRACT: Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre.
05/2005;
