We prove a cabling formula for the concordance invariant $ u^+$, defined by the author and Hom. This gives rise to a simple and effective 4-ball genus bound for many cable knots.
Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.
Let $K$ be a non-trivial knot in $S^3$, and let $r$ and $r$ be two distinct rational numbers of same sign, allowing $r$ to be infinite; we prove that there is no orientation-preserving homeomorphism between the manifolds $S^3_r(K)$ and $S^3_{r}(K)$.
We further generalize this uniqueness result to knots in arbitrary integral homology L-spaces.
In this paper, we write down a special Heegaard diagram for a given product three manifold $Sigma_gtimes S^1$. We use the diagram to compute its perturbed Heegaard Floer homology.
