Title:

On some properties of the sutured Floer polytope

Each chapter of this thesis is a selfcontained article on sutured Floer homology. Chapter 1: This article is a purely expository introduction to sutured Floer homology for graduate students in geometry and topology. The article contains most of the things the author wishes she had known when she started her journey into the world of sutured Floer homology. It is divided into three parts. The first part is an introductory level exposition of Lagrangian Floer homology. The second part is a construction of Heegaard Floer homology as a special, and slightly modified, case of Lagrangian Floer homology. The third part covers the background on sutured manifolds, the definition of sutured Floer homology, as well as a discussion of its most basic properties and implications (it detects the product, behaves nicely under surface decompositions, defines an asymmetric polytope, its Euler characteristic is computable using Fox calculus). Chapter 2: We exhibit the first example of a knot K in the threesphere 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 Spincgrading. This answers a question of Juh´asz. 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 Juh´asz. Actually, we exhibit an infinite family of knots with pairs of Seifert surfaces that can be distinguished by the Euler characteristic. Chapter 3: For closed 3manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsv´ath and Szab´o, Ni, and Hedden. For example, given a closed 3manifold 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 dHF(Y ). We prove a similar bijection and duality result for a class of 3manifolds with boundary called sutured manifolds. A sutured manifold is essentially a cobordism between two possibly disconnected surfaces with boundary R+ and R−. 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 Juh´asz, 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.
