The reduced knot Floer complex

Abstract

We define a “reduced” version of the knot Floer complex CFK−(K), and show that it behaves well under connected sums and retains enough information to compute Heegaard Floer d-invariants of manifolds arising as surgeries on the knot K. As an application to connected sums, we prove that if a knot in the three-sphere admits an L-space surgery, it must be a prime knot. As an application to the computation of d-invariants, we show that the Alexander polynomial is a concordance invariant within the class of L-space knots, and show the four-genus bound given by the d-invariant of +1-surgery is independent of the genus bounds given by the Ozsváth–Szabó τ invariant, the knot signature and the Rasmussen s invariant.