On sutured Floer homology and the equivalence of Seifert surfaces

Abstract

The goal of this paper is twofold. First, given a Seifert surface R in the 3-sphere, we show how to construct a Heegaard diagram for the sutured manifold S3(R) complementary to R, which in turn enables us to compute the sutured Floer homology of S3(R) combinatorially. Secondly, we outline how the sutured Floer homology of S3(R), together with the Seifert form of R, can be used to decide whether two minimal genus Seifert surfaces of a given knot are isotopic in S3. We illustrate our techniques by showing that the knot 83 has two minimal genus Seifert surfaces up to isotopy. Furthermore, for any n ≥ 1 we exhibit a knot Kn that has at least n nonisotopic free minimal genus Seifert surfaces.