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$τ$-invariants for knots in rational homology spheres

Ozsváth and Szabó used the knot filtration on $\widehat{CF}(S^3)$ to define the $τ$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $τ$-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 $Y$..