Using an invariant knot of a flow to find additional invariant structure

Abstract

Consider a continuous flow in $\mathbb{R}^3$ or any orientable $3$-manifold. Let $(Q_1, Q_0)$ be an index pair in the sense of Conley and consider the region $N := \overline{Q_1 - Q_0}$. (An example of this is a compact $3$-manifold $N$ such that trajectories of the flow cross $\partial N$ inwards or outwards transversally, or bounce off it from the outside). Suppose we know there is an invariant knot or link $K$ in the interior of $N$. We prove the following: if $K$ is contractible and nontrivial (in the sense of knot theory) in $N$, then every neighbourhood $U$ of $K$ contains a point $p \in N - K$ such that the whole trajectory of $p$ is contained in $N$. In other words, the presence of $K$ forces the existence of additional invariant structure in $N$ (besides $K$), and the latter can actually be found arbitrarily close to $K$. To prove this result we develop a ``coloured'' handle theory which may be of independent interest to study flows in $3$-manifolds.