Knot-theoretical invariants for 3-dimensional vector fields

Abstract:

In some physical contexts (hydro or magneto dynamics) it is natural to consider some 3-dimensional volume-preserving vector fields up to volume-preserving diffeomorphism. In order to understand the classes of such objects, it is then a natural goal to construct invariants. The possible existence of periodic orbits and the number of those is such an invariant. The knot type of those is a more precise one. However periodic orbits may be hard to detect and give some information that may be considered too local or discrete. In this survey talk, we will show how some constructions of knot-theory can be exported to this context and yield some "continuous" invariants, in particular the so-called helicity.