Floer homology and surface decompositions

Abstract

Sutured Floer homology, denoted by $SFH$, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if $\left(M,\gamma \right)⇝\left(M^{\prime },{\gamma }^{\prime }\right)$ is a sutured manifold decomposition then $SFH\left(M^{\prime },{\gamma }^{\prime }\right)$ is a direct summand of $SFH\left(M,\gamma \right)$. To prove the decomposition formula we give an algorithm that computes $SFH\left(M,\gamma \right)$ from a balanced diagram defining $\left(M,\gamma \right)$ that generalizes the algorithm of Sarkar and Wang.

As a corollary we obtain that if $\left(M,\gamma \right)$ is taut then $SFH\left(M,\gamma \right)\ne 0$. Other applications include simple proofs of a result of Ozsváth and Szabó that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry.

Moreover, using these methods we show that if $K$ is a genus $g$ knot in a rational homology $3$–sphere $Y$ whose Alexander polynomial has leading coefficient $a_g\ne 0$ and if $\text{ rk}\widehat{HFK}\left(Y,K,g\right)<4$ then $Y\setminus N\left(K\right)$ admits a depth $\le 2$ taut foliation transversal to $\partial N\left(K\right)$.