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.
Categories: math.GT