# A surgery formula for knot Floer homology

## TL;DR

This paper refines the surgery formula for knot Floer homology by introducing a second filtration that computes the dual knot's Floer complex, along with a new combinatorial Alexander grading formula.

## Contribution

It enhances the Ozsváth-Szabó surgery formula by adding a second filtration for dual knots and provides a new combinatorial Alexander grading formula.

## Key findings

- A refined surgery formula with a second filtration for dual knots.
- A combinatorial Alexander grading formula of independent interest.
- Improved understanding of knot Floer homology under surgery.

## Abstract

Let $K$ be a rationally null-homologous knot in a $3$-manifold $Y$, equipped with a nonzero framing $\lambda$, and let $Y_\lambda(K)$ denote the result of $\lambda$-framed surgery on $Y$. Ozsv\'ath and Szab\'o 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.