On invariants for Legendrian knots

András I. Stipsicz and Vera Vértesi

Vol. 239 (2009), No. 1, 157–177

DOI: 10.2140/pjm.2009.239.157

Abstract

Let (Y,ξ) be a contact 3-manifold and L a null-homologous Legendrian knot in it. We determine the connection between the sutured invariant EH(L) = EH(Y −ν(L),ξ|_{Y −ν(L)}) of L and the Legendrian invariant ℒ(L) defined in a paper by Lisca, Ozsváth, Stipsicz and Szabó. We derive a vanishing theorem for ℒ(L) in the presence of Giroux torsion in the complement of the knot, and reprove several known properties of the Legendrian invariant from this perspective.

Keywords

Legendrian and transverse knot, Heegaard Floer homology, sutured Floer homology

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 53C15

Milestones

Received: 9 June 2008

Accepted: 24 July 2008

Published: 1 January 2009