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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 (M,γ)→(M',γ')
is a sutured manifold decomposition then SFH(M',γ') is a direct
summand of SFH(M,γ). To prove the decomposition formula we give
an algorithm that computes SFH(M,γ) from a balanced diagram defining
(M,γ) that generalizes the algorithm of Sarkar and Wang.
As a corollary we obtain that if (M,γ) is taut then
SFH(M,γ)≠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 ag≠0 and if rk(^HFK(Y,K,g))<4
then Y╲N(K) admits a depth ≤2 taut foliation transversal
to ∂N(K).
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