On transverse invariants from Khovanov homology
type: Item
instance of: scholarly article
title: On transverse invariants from Khovanov homology
zbMATH Open document ID: 1327.57010
DOI: 10.4171/QT/69
author: Robert Lipshitz
author: Sucharit Sarkar
author: Lenhard Ng
published in: Quantum Topology
publication date: 2015-11-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.6371
review text: Summary: In [Math. Res. Lett. 13, No. 4, 571--586 (2006; Zbl 1143.57006)], \textit{O. Plamenevskaya} associated to each transverse knot \(K\) an element of the Khovanov homology of \(K\). In this paper, we give two refinements of Plamenevskaya's invariant, one valued in \textit{D. Bar-Natan}'s deformation (from [Geom. Topol. 9, 1443--1499 (2005; Zbl 1084.57011)]) of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum (from [\textit{R. Lipshitz} and \textit{S. Sarkar}, J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)]). We show that the first of these refinements is invariant under negative flypes and \(SZ\) moves; this implies that Plamenevskaya's class is also invariant under these moves. We go on to show that for small-crossing transverse knots \(K\), both refinements are determined by the classical invariants of \(K\).
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R17
zbMATH DE Number: 6505363
zbMATH Keywords: transverse knot
zbMATH Keywords: Khovanov homology
zbMATH Keywords: flype
zbMATH Keywords: Khovanov spectrum
MaRDI profile type: Publication
arXiv ID: 1303.6371
cites work: On the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology
cites work: Khovanov's homology for tangles and cobordisms
cites work: Q3692516
cites work: Studying links via closed braids. III: Classifying links which are closed 3-braids
cites work: Stabilization in the braid groups. I: MTWS
cites work: Stabilization in the braid groups. II. Transversal simplicity of knots
cites work: Differential algebra of Legendrian links
cites work: Q4850928
cites work: Filtrations on the knot contact homology of transverse knots
cites work: Chekanov-Eliashberg invariants and transverse approximations of Legendrian knots.
cites work: Q3378435
cites work: Cabling and transverse simplicity
cites work: Legendrian and transverse cables of positive torus knots
cites work: Q4459930
cites work: Legendrian and transverse twist knots
cites work: A family of transversely nonsimple knots
cites work: An invariant of tangle cobordisms
cites work: Link homology and Frobenius extensions
cites work: An endomorphism of the Khovanov invariant
cites work: A Khovanov stable homotopy type
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description: scientific article