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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 homologyinvariants of the sutured manifolds obtained by cutting the knot complementsalong the Seifert surfaces. Our examples provide the first use of sutured Floerhomology, and not merely its Euler characteristic(a classical torsion), todistinguish Seifert surfaces. Our technique uses a version of Floer homology,called Longitude Floer Homology in a way that enables us to bypass thecomputations related to the SFH of the complement of a Seifert surface.

Seifert surfaces distinguished by sutured Floer homology but not its Euler characteristic