Heegaard Floer homology of surgeries on two-bridge links
type: Item
instance of: scholarly article
title: Heegaard Floer homology of surgeries on two-bridge links
zbMATH Open document ID: 1505.57020
published in: Algebraic \& Geometric Topology
publication date: 2022-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.5727
review text: \textit{C. Manolescu} and \textit{P. Ozsváth} [``Heegaard Floer homology and integer surgeries on links'', Preprint, \url{arXiv:1011.1317}] developed a formula for the Heegaard Floer homology of surgeries on links. The author uses this formula to give a practical polynomial time algorithm to compute the Floer homology of surgeries on 2-bridge links. The algorithm mainly utilizes genus-0 nice diagrams and algebraic arguments to simplify the Manolescu-Ozsváth formula. In fact, two sets of data are needed in the surgery formula: the generalized Floer complex \(A^-_s(\vec{L})\) for a link \(L\), and the maps \(\mathcal{I}^{\vec{L'}}_s\) and \(D^{\vec{L'}}_s\) connecting the complexes associated to oriented sublinks \(L' \subset L\). Here is what the author does to simplify the Manolescu-Ozsváth formula: 1. The Schubert normal form of 2-bridge links is utilized to get a nice Heegaard diagram. With the nice diagram, the Floer complex \(A^-_s(\vec{L})\) can be directly computed. 2. The author defines a twisted gluing of squares of chain complexes. Then the link surgery formula is a twisted gluing of certain squares of chain complexes derived from the link \(L\). 3. The author shows some rigidity results of maps \(D^{\vec{L'}}_s\) up to chain homotopy for 2-bridge links, which mean perturbing the maps \(D^{\pm L'}_s\) by chain homotopy does not change the isomorphism type of the original complex. The author also shows that \(\boldsymbol{HF}^-\) and \(d\)-invariants of surgeries on a 2-bridge link \(L\) are determined only by the homotopy type of \(A^-_s(\vec{L})\). Some examples are computed in this paper, which provide new examples of hyperbolic 3-manifolds whose Heegaard Floer homology can be computed.
reviewed by: Jingling Yang
Mathematics Subject Classification ID: 57K18
Mathematics Subject Classification ID: 57K10
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 7570600
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: link surgery formula
zbMATH Keywords: 2-bridge links
author: Ya-Jing Liu
MaRDI profile type: Publication
OpenAlex ID: W1550719593
Wikidata QID: Q114045612
arXiv ID: 1402.5727
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DOI: 10.2140/AGT.2022.22.473
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description: scientific article