  We define a torsion invariant T for every balanced sutured manifold (M,g),
and show that it agrees with the Euler characteristic of sutured Floer homology
SFH. The invariant T is easily computed using Fox calculus. With the help of T,
we prove that if (M,g) is complementary to a Seifert surface of an alternating
knot, then SFH(M,g) is either 0 or Z in every spin^c structure. T can also be
used to show that a sutured manifold is not disk decomposable, and to
distinguish between Seifert surfaces.
  The support of SFH gives rise to a norm z on H_2(M, \partial M; R). Then T
gives a lower bound on the norm z, which in turn is at most the sutured
Thurston norm x^s. For closed three-manifolds, it is well known that Floer
homology determines the Thurston norm, but we show that z < x^s can happen in
general. Finally, we compute T for several wide classes of sutured manifolds.
