A surgery formula for knot Floer homology

Abstract

Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a non-zero framing $\lambda$, and let $Y_{\lambda }(K)$ denote the result of $\lambda$-framed surgery on $Y$. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of $Y_{\lambda }(K)$ in terms of the knot Floer complex of $(Y,K)$. We strengthen this formula by adding a second filtration that computes the knot Floer complex of the dual knot $K_{\lambda }$ in $Y_{\lambda }$, i.e., the core circle of the surgery solid torus. In the course of proving our refinement we derive a combinatorial formula for the Alexander grading which may be of independent interest.