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

Consider a continuous flow in R³ or any orientable $3$-manifold. Let (Q₁, Q₀) be an index pair in the sense of Conley and consider the region N := Q₁ - Q₀̄. (An example of this is a compact $3$-manifold N such that trajectories of the flow cross ∂ 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 ∈ 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.