The next-to-top term in knot Floer homology

Abstract

Let $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)\le 1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $\widehat{\mathrm{HFK}}(Z,K,[F],g)$ is supported in a single $\mathbb{Z}/2\mathbb{Z}$-grading, then