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

TL;DR

The paper demonstrates that an invariant knot within a flow in a 3-manifold implies the existence of additional invariant trajectories nearby, using a novel coloured handle theory approach.

Contribution

It introduces a new coloured handle theory to analyze flows in 3-manifolds and shows how invariant knots induce further invariant structures.

Findings

Invariant knots imply nearby invariant trajectories.

The approach applies to flows in $\\mathbb{R}^3$ and orientable 3-manifolds.

Develops a new coloured handle theory for flow analysis.

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.