Sutured Floer homology and invariants of Legendrian and transverse knots

TL;DR

This paper introduces a new contact-geometric approach to sutured Floer homology, defining invariants for Legendrian and transverse knots that align with existing invariants, and provides computational methods using bordered sutured Floer homology.

Contribution

It offers an alternative formulation of knot Floer homology using contact geometry and sutured Floer homology, and constructs new invariants for Legendrian and transverse knots that match known invariants.

Findings

New invariants EHL and EHIL for Legendrian and transverse knots

Invariants agree with previously defined invariants despite different definitions

Explicit computations facilitated by bordered sutured Floer homology

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,\xi), 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.