Knot Floer homology and Seifert surfaces

Abstract

Let $K$ be a knot in $S^3$ of genus $g$ and let $n>0.$ We show that if $rk\widehat{HFK}\left(K,g\right)<2^{n+1}$ (where $\widehat{HFK}$ denotes knot Floer homology), in particular if $K$ is an alternating knot such that the leading coefficient $a_g$ of its Alexander polynomial satisfies $\left|a_g\right|<2^{n+1},$ then $K$ has at most $n$ pairwise disjoint nonisotopic genus $g$ Seifert surfaces. For $n=1$ this implies that $K$ has a unique minimal genus Seifert surface up to isotopy.