Sutured Floer homology and invariants of Legendrian and transverse knots

Abstract

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,ξ), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. The distinguished class EHL is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsvath, Stipsicz, and Szabo despite a strikingly dissimilar definition. While our definitions and constructions only involve sutured Floer homology and contact geometry, the identification of our invariants with known invariants uses bordered sutured Floer homology to make explicit computations of maps between sutured Floer homology groups.

100 pages, 50 figures

References in corpus (10)

• Floer homology and knot complements
• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Right-veering diffeomorphisms of compact surfaces with boundary II
• Lectures on open book decompositions and contact structures
• Contact structures, sutured Floer homology and TQFT
• Bordered Floer homology for sutured manifolds
• The vanishing of the contact invariant in the presence of torsion
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Instanton Floer homology and contact structures
• Joining and gluing sutured Floer homology

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• The cosmetic crossing conjecture for split links
• Functoriality of the EH class and the LOSS invariant under Lagrangian concordances
• Non-loose negative torus knots
• Strand algebras and contact categories
• On contact surgery and knot Floer invariants
• Formal Contact Categories
• On computation of morphism spaces and a direct limit of the bordered Floer homology of knot complements
• 2-torsion in instanton Floer homology