The work of Donaldson and Mark made the structure of the SeibergWitten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. In the paper, these invariants are studied by using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and detailed computations are made for knot complements.