Sutured Floer homology distinguishes between Seifert surfaces

Abstract

We exhibit the first example of a knot K in the three-sphere with a pair of minimal genus Seifert surfaces R1 and R2 that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spinc grading. This answers a question of Juhász. More precisely, we show that the Euler characteristic of the sutured Floer homology distinguishes between R1 and R2, as does the sutured Floer polytope introduced by Juhász. Actually, we exhibit an infinite family of knots with pairs of Seifert surfaces that can be distinguished by the Euler characteristic.