A surgery formula for knot Floer homology

Abstract

Let K be a rationally null-homologous knot in a 3-manifold Y , equipped with a nonzero framing λ, and let Y λ (K) denote the result of λ-framed surgery on Y . Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Y λ (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 λ in Y λ , 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.

Knot Floer homology and relative adjunction inequalities

“…Second, there are different conventions for the definition of the Alexander grading in the literature [36,37,18,16]. Ours is consistent with [16].…”

“…This article is, in a sense, the sequel of [13]. Here, however, we use the more general construction of knot Floer homology for rationally null-homologous knots, and in the first subsection we recall and clarify the structure of the theory in this setting; see [16] for further details. Having done this, we turn to the definition and elementary properties of the generalized τ invariants, which we collect in Subsection 2.2.…”

Involutive Heegaard Floer homology

“…The equivalence with other formulations is verified in [HL19a,Lemma 2.10]. An alternate approach for null-homologous knots may be found in [Zem19b, Theorem 2.13 (e); Proposition 8.1]).…”

“…Any other Spin c structure with the same boundary restrictions may be written as u i,j = (w 0 + iPD[ Σd ])#(z 0 + jPD[S]), boundary is −K, so that one can cap off the knot cobordism to obtain a class H 2 (W ; Z). In [HL19a], the authors define the Alexander grading using a Seifert surface whose boundary is K.…”

Contact (+1)-surgeries on rational homology 3-spheres

“…Here, we use the convention in [22]. We also denote it by CF K ∞ (Y, K, s), and abbreviate it by C s .…”