Sutured Floer homology distinguishes between Seifert surfaces

Abstract

We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This answers a question of Juh\'asz. More precisely, we show that the Euler characteristic of the sutured Floer homology of the complementary manifolds distinguishes between the two surfaces, as does the sutured Floer polytope introduced by Juh\'asz. Actually, we exhibit an infinite family of knots with pairs of Seifert surfaces that can be distinguished by the Euler characteristic.