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

TL;DR

This paper explores the relationship between Morse-Novikov theory, Heegaard splittings, and the counting of closed orbits in gradient flows on 3-manifolds, providing detailed computations for knot complements.

Contribution

It introduces new methods combining Morse-Novikov theory and Heegaard splittings to compute torsion invariants related to gradient flows on 3-manifolds.

Findings

Computed torsion invariants for knot complements

Linked Seiberg-Witten invariants with flow line counts

Enhanced understanding of gradient flow structures on 3-manifolds

Abstract

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.