Knotting Minimal Sets

2025/09/21 by Alex S. Clark, Clark, Alex, John Hunton +1
Computer Science · Engineering · #37B35 #37B52 (Secondary) #37C10 #57K12 (Primary) 37B10 #Architecture and Computational Design #Constraint Satisfaction and Optimization #Design Education and Practice #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.17133

openalex publication_date 2025/09/21 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We consider the ways minimal sets of flows in S³ may be embedded. We prove that given any C² flow on S³ with positive entropy, there is an uncountable collection M of topologically distinct minimal sets such that for each M∈ M there are infinitely many embedded copies of M in the flow, each copy with a distinct knot type, thus extending work of Franks and Williams for periodic orbits.

Citations

• A Complete Invariant for Flow Equivalence
• Knotted toroidal sets, attractors and incompressible surfaces
• The dynamics of generic Kuperberg flows
• 3-manifolds that admit knotted solenoids as attractors

Related

• Graph representations of surface flows
• Some properties of simple minimal knots
• Topological entropy and orbit growth in link complements
• Flows with minimal subdynamics
• Almost proximal extensions of minimal flows