$τ$-invariants for knots in rational homology spheres

Katherine Raoux

Published 2016-11-28Version 1

Ozsv\'ath and Szab\'o used the knot filtration on $\hat{CF}(S^3)$ to define the $\tau$-invariant for knots in $S^3$. In this article we generalize their construction and define a collection of invariants $\{\tau_\mathfrak{s}(Y,K)\}_{\mathfrak{s}\in\S(Y)}$ for rationally null-homologous knots in rational homology spheres. We also show that these invariants can be used to obtain a lower bound on the genus of a surface with boundary $K$ properly embedded in a negative definite 4-manifold with boundary $Y$.