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 versions 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 $K\subset \left(Y,\xi \right)$ we associate distinguished classes $\underrightarrow{EH}\left(K\right)\in {HFK}^{-}\left(-Y,K\right)$ and $\underleftarrow{EH}\left(K\right)\in {HFK}^{+}\left(-Y,K\right)$, which are each invariant under Legendrian or transverse isotopies of $K$. The distinguished class $\underrightarrow{EH}$ is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsváth, Stipsicz and Szabó 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.