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 of 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 Ozsv\'ath-Szab\'o $\tau$ invariant, the knot signature and the Rasmussen $s$ invariant.Comment: 41 pages, 14 figures; changed formatting, updated references, added some clarifying remarks, results unchange

Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link

“…Given an L-space knot K, let C = CF K ∞ (K){i ≤ 0}. Then the reduced complex CF K − (K) consists merely of the generators in C which have no outgoing nor incoming horizontal arrows, and has trivial differential (see [Krc15,Corollary 4.2]). As a complex, CF K − (K) ∼ = F[U ], with 1 supported in grading zero.…”

“…For the precise statement of vanishing Hedden-Livingston-Ruberman obstruction, see Theorem 2.8. To prove that Σ K Ln has non-vanishing Hedden-Livingston-Ruberman obstruction, we will need the following theorem which is based on results of [KP16] and the reduced Floer chain complex introduced by the second author [Krc15]. Here, {V k (K)} is a sequence of smooth concordance invariants of K which was introduced by Rasmussen [Ras03] and then further studied by Ni and Wu [NW15].…”

Correction Terms and the Nonorientable Slice Genus

“…One computes σ(K 1 ) = −2, σ(K 2 ) = −16, υ(K 1 ) = −1 and υ(K 2 ) = −6. Using the techniques from [13] as in [5], we can also compute ϕ(K) = 6 and ϕ(K) = 0.…”

Heegaard Floer Homologies and Rational Cuspidal Curves. Lecture notes.

“…28 (a) L-space knots are prime. A connected sum of two non-trivial knots is never an L-space knot (see [40]). [92,Proposition 9.6] and [29]).…”