Classification of nonsingular four-dimensional flows with a untwisted saddle orbit

The topological equivalence of low-dimensional Morse–Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse–Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$.
Bibliography: 24 titles.