Knot Floer homology and Seifert surfaces

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Let K be a knot in S³ of genus g and let n>0. We show that if rk HFK(K,g) < 2ⁿ⁺¹ (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient ag of its Alexander polynomial satisfies |ag| <2ⁿ⁺¹, then K has at most n pairwise disjoint non-isotopic genus g Seifert surfaces. For n=1 this implies that K has a unique minimal genus Seifert surface up to isotopy.