Nearly fibered links with genus one

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 the rank of their link Floer homology group \(\widehat{HFL}\) in the maximal (collapsed) Alexander grading \(s_{\text{top}}\) is equal to two. In other words, such a link \(L\) satisfies \(s_{\text{top}}=\frac{n-\chi(\Sigma)}{2}=1\), and in addition \({\rm rk}\widehat{HFL}_{*}(L)[1]=2\) and \({\rm rk}\widehat{HFL}_{*}(L)[s]=0\) 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.