Dynamical Systems on Three Manifolds Part I: Knots, Links and Chaos

TL;DR

This paper constructs explicit dynamical systems on a solid torus that can contain any knot or link, including chaotic systems modeled after the Smale horseshoe, advancing the understanding of knot dynamics and chaos.

Contribution

It provides a method to embed any knot or link into a dynamical system on a torus, including chaotic systems, using braid representations and explicit differential equations.

Findings

Constructed systems containing arbitrary knots and links.

Embedded chaotic dynamics via Smale horseshoe modeling.

Explicit differential equations for knotted trajectories.

Abstract

In this paper, we give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tube neigbourhood of the braid. The second one, generating chaotic systems, is realized by modeling the Smale horseshoe.