Comparing invariants of Legendrian knots
type: Item
instance of: scholarly article
title: Comparing invariants of Legendrian knots
zbMATH Open document ID: 1334.57009
DOI: 10.4171/QT/66
author: Marco Golla
published in: Quantum Topology
publication date: 2015-11-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.1203
review text: To a Legendrian knot \(L\) in a closed contact \(3\)-manifold \(Y\) one associates a contact \(3\)-manifold with convex boundary \(Y_L\) defined as the complement of a standard neighborhood of \(L\) in \(Y\). The contact invariant of \(Y_L\), defined by Honda, Kazez and Matić [\textit{K. Honda} et al., Invent. Math. 176, No. 3, 637--676 (2009; Zbl 1171.57031)] and denoted by \(EH(L)\), can be seen as an invariant of the Legendrian isotopy class of \(L\). In addition, Lisca, Ozsváth, Stipsicz and Szabó in [\textit{P. Lisca} et al., J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307--1363 (2009; Zbl 1232.57017)] using open book decompositions constructed an invariant of an oriented nullhomologous Legendrian knot \(L\) in a closed contact 3-manifold \((Y,\xi)\) which is called \({\mathfrak L}^{-}(L)\). The author proves the equivalence of the invariants \(EH(L)\) and \({\mathfrak L}^{-}(\pm L)\) for oriented Legendrian knots \(L\) in \((S^3, \xi_{st})\), where \(\xi_{st}\) is the standard contact structure on \(S^3\). Here \(-L\) denotes \(L\) with reversed orientation. More precisely, given two oriented, topologically isotopic Legendrian knots \(L_0\), \(L_1\) in \((S^3,\xi_{st})\), the author proves that the following two statements are equivalent: \(EH(L_0)=EH(L_1)\) and \({\mathfrak L^{-}}(\pm L_0) = {\mathfrak L}^{-}(\pm L_1)\).
reviewed by: Roman Golovko
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 6505360
zbMATH Keywords: Legendrian and transverse knot
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: sutured Floer homology
MaRDI profile type: Publication
arXiv ID: 1404.1203
cites work: On the equivalence of Legendrian and transverse invariants in knot Floer homology
cites work: Contact 3-manifolds twenty years since J. Martinet's work
cites work: Topologically trivial Legendrian knots
cites work: Q3414429
cites work: Knots and contact geometry. I: Torus knots and the figure eight knot.
cites work: Fibered Transverse Knots and the Bennequin Bound
cites work: Sutured Floer homology and invariants of Legendrian and transverse knots
cites work: Q4549294
cites work: Ozsváth-Szabó invariants of contact surgeries
cites work: Knot Floer homology of Whitehead doubles
cites work: NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT
cites work: Dehn surgery, rational open books and knot floer homology
cites work: On the classification of tight contact structures. I
cites work: The contact invariant in sutured Floer homology
cites work: Holomorphic discs and sutured manifolds
cites work: A cylindrical reformulation of Heegaard Floer homology
OpenAlex ID: W2963939576
description: scientific article