  In this paper we find a family of knots with trivial Alexander polynomial,
and construct two non-isotopic Seifert surfaces for each member in our family.
In order to distinguish the surfaces we study the sutured Floer homology
invariants of the sutured manifolds obtained by cutting the knot complements
along the Seifert surfaces. Our examples provide the first use of sutured Floer
homology, and not merely its Euler characteristic(a classical torsion), to
distinguish Seifert surfaces. Our technique uses a version of Floer homology,
called Longitude Floer Homology in a way that enables us to bypass the
computations related to the SFH of the complement of a Seifert surface.
