The study of fluid dynamics on Riemannian manifolds has traditionally drawn upon tools from analysis and, more recently, geometry. By using ideas and techniques from the topology of contact structures, we will introduce topological methods for analyzing and creating new examples of solutions to the Euler equations which are independent of any geometric structure. We then couple these topological techniques with the "template theory" of Birman and Williams to ascertain results about knotted flowlines in 3-d fluid flows.