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

The works of Donaldson and Mark make 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. We study these invariants using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements.Comment: 27 pages, 12 figure

Complements of hyperbolic knots of braid index four contain no closed embedded totally geodesic surfaces

AbstractWe prove that if F is a closed essential surface embedded in the complement of a knot K of braid index four, then at least one of the following holds: (1) F is meridionally compressible, (2) K is isotopic to a simple closed curve on F, (3) there is an essential annulus properly embedded in the closure of the component of S3−N(F) which does not contain K. We obtain a corollary that there are no closed embedded totally geodesic surfaces in the complements of hyperbolic knots of braid index four