Dehn twists in Heegaard Floer homology
type: Item
instance of: scholarly article
title: Dehn twists in Heegaard Floer homology
zbMATH Open document ID: 1209.57017
author: Bijan Sahamie
published in: Algebraic \& Geometric Topology
publication date: 2010-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.3204
review text: The main results of this paper are two long exact sequences in the hat-version of Heegaard Floer homology in \(\mathbb{Z}_2\)-coefficients, together with their application in contact geometry. A Heegaard splitting represents a \(3\)-manifold \(Y\) as glued from two handlebodies of genus \(g\) along their boundary \(\Sigma\). A Heegaard diagram describes the gluing map using two collections of \(g\) embedded curves \((\alpha, \beta)\). The collections \(\alpha\) and \(\beta\) define Lagrangian tori \(T_\alpha\) and \(T_\beta\) in the \(g\)-fold symmetric product \(Sym^g \Sigma\). When the Heegaard diagram is admissible, the hat-version of Heegaard Floer homology \(\widehat{HF}(Y)\) is defined from a variant of the Lagrangian Floer homology for the pair \(T_\alpha\) and \(T_\beta\) [\textit{P.~Ozsváth} and \textit{Z.~Szabó}, Ann. Math. (2) 159, 1027--1158 (2004; Zbl 1073.57009)]. Let \(K\) be a knot in \(Y\), then a Heegaard splitting can be chosen such that \(K\) is an embedded curve on \(\Sigma\). Using a Heegaard diagram subordinate to the knot \(K\), a similar variant of the Lagrangian Floer homology defines the hat-version of knot Floer homology \(\widehat{HFK}(Y, K)\). In \S 2, the author shows that the construction of \(\widehat{HFK}(Y, K)\) [\textit{P.~Ozsváth} and \textit{Z.~Szabó}, Adv. Math. 186, 58--116 (2004; Zbl 1062.57019)] extends to arbitrary knots \(K\). Let \(Y_0(K)\) (respectively \(Y_{\pm 1}(K)\)) be the result of \(0\)-surgery (respectively \((\pm 1)\)-surgery) along the knot \(K\) and let \(\mu\) be the meridian of \(K\) in \(Y\). The main result of the present paper is the exact sequence proven in \S 4: \[ \ldots \to \widehat{HFK}(Y, K) \to \widehat{HF}(Y_{-1}(K)) \to \widehat{HFK}(Y_0(K), \mu) \to \ldots \] and a similar one with \(\widehat{HF}(Y_{+1}(K))\) and opposite arrows instead. The Dehn twists in the title is due to the fact that \(Y_{\pm 1}(K)\) can be obtained by performing a \((\mp)\)-Dehn surgery along \(K\) on the gluing map that …
reviewed by: Shengda Hu
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5705453
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: contact geometry
zbMATH Keywords: contact element
zbMATH Keywords: Legendrian knot
MaRDI profile type: Publication
OpenAlex ID: W2020646289
arXiv ID: 0904.3204
cites work: Q3136738
cites work: Symplectic fillability of tight contact structures on torus bundles
cites work: A Legendrian surgery presentation of contact 3-manifolds
cites work: Handle moves in contact surgery diagrams
cites work: Q4821921
cites work: Lutz twist and contact surgeries
cites work: Classification of overtwisted contact structures on 3-manifolds
cites work: TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2
cites work: Q3414429
cites work: An Introduction to Contact Topology
cites work: Q4700561
cites work: The contact invariant in sutured Floer homology
cites work: On the contact class in Heegaard Floer homology
cites work: Holomorphic discs and sutured manifolds
cites work: Heegard Floer invariants of Legendrian knots in contact three-manifolds
cites work: Ozsváth-Szabó invariants and tight contact three-manifolds; I
cites work: Notes on the contact Ozsváth-Szabó invariants
cites work: Ozsváth-Szabó invariants and tight contact three-manifolds. II
cites work: Contact surgeries and the transverse invariant in knot Floer homology
cites work: Heegaard Floer homology and alternating knots
cites work: Holomorphic disks and knot invariants
cites work: Holomorphic disks and three-manifold invariants: properties and applications
cites work: Holomorphic disks and topological invariants for closed three-manifolds
cites work: Heegaard Floer homology and contact structures
cites work: On invariants for Legendrian knots
DOI: 10.2140/AGT.2010.10.465
description: scientific article