Abstract. We exhibit the first example of a knot K in the three-sphere with a pair of minimal genus Seifert surfaces R 1 and R 2 that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin c -grading. This answers a question of Juhász. More precisely, we show that the Euler characteristic of the sutured Floer homology distinguishes between R 1 and R 2 , 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.

Abstract. For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y , there is a bijection between vertices of the HF + (Y ) polytope carrying the group Z and the faces of the Thurston norm unit ball that correspond to fibrations of Y over the unit circle. Moreover, the Thurston norm unit ball of Y is dual to the polytope of HF (Y ).We prove a similar bijection and duality result for a class of 3-manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two surfaces R+ and R− that have nonempty boundary. We show that there is a bijection between vertices of the sutured Floer polytope carrying the group Z and equivalence classes of taut depth one foliations that form the foliation cones of Cantwell and Conlon. Moreover, we show that a function defined by Juhász, which we call the geometric sutured function, is analogous to the Thurston norm in this context. In some cases, this function is an asymmetric norm and our duality result is that appropriate faces of this norm's unit ball subtend the foliation cones.An important step in our work is the following fact: a sutured manifold admits a fibration or a taut depth one foliation whose sole compact leaves are exactly the connected components of R+ and R−, if and only if, there is a surface decomposition of the sutured manifold resulting in a connected product manifold.

Abstract. For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology (SFH ) can be used to determine all fibred classes in H 1 (M ). Furthermore, we show that the SFH of a balanced sutured manifold (M, γ) detects which classes in H 1 (M ) admit a taut depth one foliation such that the only compact leaves are the components of R(γ). The latter had been proved earlier by the first author under the extra assumption that H 2 (M ) = 0. The main technical result is that we can obtain an extremal Spin c -structure s (i.e., one that is in a 'corner' of the support of SFH ) via a nice and taut sutured manifold decomposition even when H 2 (M ) = 0, assuming the corresponding group SF H(M, γ, s) has non-trivial Euler characteristic.