Knot Floer homology and rational surgeries
type: Item
instance of: scholarly article
title: Knot Floer homology and rational surgeries
zbMATH Open document ID: 1226.57044
author: Peter Ozsváth
author: Zoltán Szabó
published in: Algebraic \& Geometric Topology
publication date: 2011-01-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0504404
review text: The article constructs a version of knot Floer homology in the context of \(K\) being a rationally null-homologous knot in a three-manifold \(Y\). Also, the Floer homology of a three-manifold obtained by Morse surgery (integer slope) on \(K\) is described. Consider \(Y_{r}(K)\) the 3-manifold obtained by surgering \(Y\) along \(K\) with slope \(r\). The Heegaard Floer homology of rational surgeries on \(Y\) along a null-homologous knot \(K\subset Y\) is expressed in terms of a filtered homotopy type of the knot invariant for \(K\). After constructing the filtration, the first theorem proved is the following Theorem 1.1. Let \(K\subset Y\) be a null-homologous knot, and let \(p\), \(q\) be a pair of relatively prime integers. Then, for each \(i\in\mathbb{Z}_{p}\), there is a relatively graded isomorphism of groups \[ H_{*}(\mathbb{X}^{+}_{i,p/q})\simeq HF^{+}(Y_{p/q}(K),i), \] where \(\mathbb{X}^{+}_{i,p/q}\) is constructed from the filtration. Next, some applications come out of theorem 1.1: Application I -- A three manifold \(Y_{r}(K)\) is an \(L\)-space if it is a rational homology 3-sphere whose Floer homology \(HF^{+}(Y_{r}(K),s)\simeq HF^{+}(S^{3},s)\). A knot \(K\subset Y\) admits an \(L\)-space surgery if there exists \(r\) such that \(Y_{r}(K)\) is an \(L\)-space. The difference among the gradings of \(\widehat{HF}(S^{3}_{p/q}(K))\) and \(\widehat{HF}(S^{3}_{p/q}(O))\) (\(O\)=trivial knot) is expressed in terms of the Alexander polynomial coefficients of \(K\); the authors prove the following result; {\textbf{ Theorem 1.2}} Let \(K\subset S^{3}\) be a knot which admits an \(L\)-space surgery, for some \(r=p/q\in\mathbb{Q}\) with \(r\geq 0\). Then, for all integers \(i\) with \(|i|\leq p/2\) we have that \[ d(S^{3}_{p/q}(K),i)-d(S^{3}_{p/q}(O,i)=-2t\mid[i/q]\mid(K), \] while for all \(|j|\geq \frac{p}{2q}\), we have that \(t_{j}(K)=0\). In Theorem 1.2, \(d\) stands for a correction term function \(d:Spin^{c}(Y)\rightarrow \mathbb{Q}\) which associates to each…
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5837746
zbMATH Keywords: Floer homology
zbMATH Keywords: Dehn surgery
reviewed by: Celso M. Doria
describes a project that uses: SnapPea
MaRDI profile type: Publication
OpenAlex ID: W3101555884
arXiv ID: math/0504404
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DOI: 10.2140/AGT.2011.11.1
description: scientific article