From Notices of the AMS
New Knots in the Lorenz Equations
by Tali Pinsky
Communicated by Daniela De Silva
In this survey, we aim to provide an overview of some of the current research and open questions concerning the relationship between flows in three-dimensional manifolds and knot theory. We will focus on a significant example: knots that arise in the Lorenz equations. Our goal is to convey the essence of the subject through intuitive explanations, while supplying references for the rigorous proofs.
Periodic Orbits as Knots
Let $M$ be a compact three-dimensional manifold. A {\it flow} on $M$ is an action of $\mathbb{R}$ on $M$ (where $\mathbb{R}$ is considered as a time coordinate) taking each point $x\in M$ at time $t$ to a point $\phi^t(x)$ so that $\phi^0(x)=x$ and $\phi^{t+s}(x)=\phi^t(\phi^s(x))$. We will assume the flowlines are continuously differentiable and thus a flow defines a vector field, and vice versa, from the existence and uniqueness of an ordinary differential equation (ODE). Since a solution to a first-order ODE can always be continued for all times on a compact manifold, any continuous vector field gives rise to a flow.
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