$\tau$–invariants for knots in rational homology spheres

Abstract

Ozsváth and Szabó used the knot filtration on $\widehat{CF}\left(S^3\right)$ to define the $\tau$–invariant for knots in the $3$–sphere. We generalize their construction and define a collection of $\tau$–invariants associated to a knot $K$ in a rational homology sphere $Y$. We then show that some of these invariants provide lower bounds for the genus of a surface with boundary $K$ properly embedded in a negative-definite $4$–manifold with boundary $Y$.