The next-to-top term in knot Floer homology

TL;DR

This paper investigates the structure of knot Floer homology for null-homologous knots in certain 3-manifolds, establishing inequalities relating the ranks of homology groups at adjacent gradings under specific conditions.

Contribution

It proves a new inequality relating the ranks of the next-to-top and top knot Floer homology groups in generalized L-spaces when supported in a single grading.

Findings

Rank of  ext{HFK}(Z,K,[F],g-1) is at least as large as  ext{HFK}(Z,K,[F],g) under certain conditions.

Supports the understanding of the structure of knot Floer homology in L-spaces.

Provides conditions under which the homology groups are supported in a single /2 grading.

Abstract

Let $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)\le1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $\widehat{HFK}(Z,K,[F],g)$ is supported in a single $\mathbb Z/2\mathbb Z$--grading, then \[\mathrm{rank}\widehat{HFK}(Z,K,[F],g-1)\ge\mathrm{rank}\widehat{HFK}(Z,K,[F],g).\]