Longitude Floer homology and the Whitehead double

Abstract

We define the longitude Floer homology of a knot $K\subset S^3$ and show that it is a topological invariant of $K$. Some basic properties of these homology groups are derived. In particular, we show that they distinguish the genus of $K$. We also make explicit computations for the $\left(2,2n+1\right)$ torus knots. Finally a correspondence between the longitude Floer homology of $K$ and the Ozsváth–Szabó Floer homology of its Whitehead double $K_L$ is obtained.