Sutured Floer homology, denoted by ,
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
is a sutured manifold
decomposition then is
a direct summand of .
To prove the decomposition formula we give an algorithm that computes
from a balanced
that generalizes the algorithm of Sarkar and Wang.
As a corollary we obtain that if
is taut then .
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
Moreover, using these methods we show that if
is a genus
knot in a rational
whose Alexander polynomial has leading coefficient
admits a depth
transversal to .