Assume we have a chaotic attractor in R^3, generated by some smooth three-dimensional flow -- can we say what makes it unique?

This question, even though intuitive, is somewhat ill posed. Let us therefore consider a better posed question: what can we say about the knot types of periodic trajectories embedded inside the attractor? This question was answered for the case when the dynamics are hyperbolic, a result often termed as the "Birman-Williams Theorem". However, most chaotic attractors are not known to be hyperbolic (i.e., "Anosov"), so in many interesting cases this question is still wide open. In this talk, we will show how using a topological invariant known as the Orbit Index (originally due to K. Alligood, J. Mallet-Paret and J.A. Yorke) one can give a partial answer to this question for two of the most famous chaotic dynamical systems: the Rössler and the Lorenz attractors. Finally (and time permitting), inspired by the "Chaotic Hypothesis" due to G. Gallavotti and by the Thurston-Nielsen Classification Theorem for surface homeomorphisms, we will discuss how our results can possibly be generalized to a larger class of three-dimensional flows.