The next-to-top term in knot Floer homology

Abstract

Let K be a null-homologous knot in a generalized L-space Z with b1(Z) ≤ 1. Let F be a Seifert surface of K with genus g. We show that if HF K(Z, K, [F ], g) is supported in a single Z/2Z-grading, then rank HF K(Z, K, [F ], g − 1) ≥ rank HF K(Z, K, [F ], g).

Homology cobordism, knot concordance, and Heegaard Floer homology

“…This seems like a hard question to answer with Heegaard Floer homology, as the question about which ι-complexes can be realized by surgery on a knot in S 3 then reduces to the question of which ι K -complexes can be realized by knots in S 3 . Even without the additional structure of ι K , this is a difficult question; for some partial answers, see [HW18], as well as more recent progress in [BVV18], [Krc15], and [Ni21].…”