2024/03/27 by J. J. Sánchez-Gabites, Sánchez-Gabites, J. J.
Computer Science · #37B35 #37E99 #57K10 #Computer Graphics and Visualization Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2403.18805
openalex publication_date 2024/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 - Q0. (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.