The decategorification of sutured Floer homology

Abstract

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.