The sutured Floer polytope and taut depth-one foliations

Abstract

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.

Sutured Floer homology, fibrations, and taut depth one foliations

“…Summary of results on foliations. We now cite two results from the first author [Al13]. We start with the following lemma that is the content of [Al13, For the reader's convenience, we give a quick outline of the proof.…”

“…The well-versed reader might be excused for having a sense of déjà vu. In fact, the theorem was proved by the first author [Al13] for strongly balanced sutured manifolds under the extra assumption that H 2 (M) = 0. This assumption was introduced for technical reasons to ensure that in the course of the proof one only works with balanced sutured manifolds.…”

Taut foliations in knot complements

“…We will state basic definitions and theorems as needed for this paper but we refer the reader to [9,10,11] for a more detailed description. The papers [1,6,19] and book [5] also provide nice descriptions of some of Gabai's sutured manifold theory. In this paper, we will use branched surfaces to describe sutured manifolds and sutured manifold decompositions.…”

“…Choose z to be the point in α 1 ∩ (∪ i β i ) that is furthest along α 1 . So there are j, t 0 , t 1 such that z = α 1 (t 0 ) = β j (t 1 ) and 1] be the concatenation of the two arcs β j [0, t 1 ] and α 1 [t 0 , 1], perturbed slightly so that it intersects α 1 transversely and minimally. For i = j, set…”

“…Corollary 4.6. There are sets of arcs A i = {α i 1 , ..., α i p }, 1 ≤ i ≤ q, such that (1) for each i, the arcs in A i are pairwise disjoint and properly embedded in…”