# Nearly fibered links with genus one

## TL;DR

This paper classifies nearly fibered links with genus one in the 3-sphere, characterized by specific properties of their link Floer homology, and computes their Floer homology groups.

## Contribution

It provides a complete classification of nearly fibered links with genus one and computes their link Floer homology groups, extending previous knot results.

## Key findings

- Classified all nearly fibered links with genus one in S^3.
- Computed the link Floer homology groups for these links.
- Connected Floer homology properties with geometric link features.

## Abstract

We classify all the $n$-component links in the $3$-sphere that bound a Thurston norm minimizing Seifert surface $\Sigma$ with Euler characteristic $\chi(\Sigma)=n-2$ and that are nearly fibered, which means that their rank of the maximal (collapsed) Alexander grading $s_{\text{top}}$ of the link Floer homology group $\widehat{HFL}$ is equal to two. In other words, such a link $L$ satisfies $s_{\text{top}}=\frac{n-\chi(\Sigma)}{2}=1$, and in addition $\text{rk}\:\widehat{HFL}_{*}(L)[1]=2$ and $\text{rk}\:\widehat{HFL}_{*}(L)[s]=0$ for every $s>1$. The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek; and involves techniques from sutured Floer homology. Furthermore, we also compute the group $\widehat{HFL}$ for each of these links.