  In this paper, we extend the theory of sutured Floer homology developed by
the author. We first prove an adjunction inequality, and then define a polytope
P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which
support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface
decomposition, then a natural map projects P(M',g') onto a face of P(M,g);
moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way
for some surface decomposition. We show that if (M,g) is reduced, horizontally
prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M;
R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most
2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In
particular, the rank of the top term of knot Floer homology bounds the
topological complexity of the knot complement, in addition to simply detecting
fibred knots.
