Morse–Novikov theory, Heegaard splittings, and closed orbits of gradient flows

The work of Donaldson and Mark made the structure of the Seiberg–Witten 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.