Holomorphic disks and knot invariants

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2002

894 citations recorded by the source corpus.

Abstract We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the Heegaard Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, and a description of the behavior under connected sums. Then, we establish a relationship with HF + for surgeries along the knot. Applications include calculation of HF + of three-manifolds obtained by surgeries on some special knots in S 3 , and also calculation of HF + for certain simple three-manifolds which fiber over the circle.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b1e14eaaf33b626efae6209b73406f66afe9744

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On knot Floer width and Turaev genus (2007, background): “Knot Floer homology is an invariant introduced by Ozsváth and Szabó [10] and independently by Rasmussen [11] that associates to each knot K S(3) a bigraded abelian group 1 HFK .”
Knot Floer homology detects fibred knots (2006, methodology): “The proof relies on previous works due to Gabai, Ozsváth–Szabó, Ghiggini and the author.”
Cables of thin knots and bordered Heegaard Floer homology (2009, methodology): “In [13], Ozsváth and Szabó introduce a powerful knot invariant using Heegaard diagrams.”
Grid Homology for Knots and Links (2015, methodology): “For additional information and updates on this book, visit www.ams.org/bookpages/surv-208 Library of Congress Cataloging-in-Publication Data Ozsváth, Peter Steven, 1967- Grid homology for knots and links / Peter S. Ozsváth, András I. Stipsicz, Zoltán Szabó. pages cm. – (Mathematical surveys and”
On combinatorial link Floer homology (2006, background): “In particular, appealing to the earlier theorem defined using holomorphic disks [9, 11, 15], the filtered quasi-isomorphism type of this chain complex C− is a link invariant.”
Bordered Floer homology for manifolds with torus boundary via immersed curves (2016, methodology): “rise to a T k i summand in spin c structure s i. Thus HF+( g ;L 0;s i ) ˘= Mg j=i+1 T ( 2g g+j ) j for each 0 <i<g+ 1. This gives HF+( g;L 0;s i) ˘=HF +(Y;s ) comparing with Ozsvath and Szabo [26]. 7.8. Splicing trefoils, revisited. As a nal example, consider the result of splicing two rightha”
Knot Floer homology and rational surgeries (2005, methodology): “In the next two sections, we turn to some properties of the knot Floer homology which are rather straightforward adaptations of the corresponding results for null-homologous knots ([22], [27]): the relationship between knot Floer homology and “large” surgeries on a rationally null-homologous knot (S”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, methodology): “For the definitions and detailed discussions of these notions, see [20]; explicit formulae will be given only in the grid context.”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, methodology): “Holomorphic disks, link invariants and the multi-variable Alexander polynomial PETER OZSVÁTH ZOLTÁN SZABÓ The knot Floer homology is an invariant of knots in S3 whose Euler characteristic is the Alexander polynomial of the knot.”
Tight contact structures on some small Seifert fibered 3-manifolds (2005, methodology): “Finally, as pointed out in [31] (see Proposition 2.1 and the proof of Lemma 4.1), by [ 29, Theorem 8.1 ] we have:”
The pillowcase and perturbations of traceless representations of knot groups (2013, background): “302] that the sutured instanton homology groups of a knot defined in [23] are isomorphic to the Heegaard knot Floer homology groups of Ozsváth-Szabó [33] and Rasmussen [35].”
Heegaard Floer homology and integer surgeries on links (2010, positive): “When L = K is a knot and the diagram H has only two basepoints (one w and one z), the complex CFL(H) is the completion of the subcomplex C{i ≤ 0} of the knot Floer complex CFK∞(Y,K), in the notation of [15].”
Knots, sutures, and excision (2008, background): “G T ] 3 1 Contents 1 Introduction 1 2 Background on monopole Floer homology 3 3 Floer’s excision theorem 19 4 Monopole Floer homology for sutured manifolds 26 5 Knot homology 33 6 Fibered knots 41 7 Instantons 52”