  The goal of this paper is to describe and clarify as much as possible the
3-dimensional topology underlying the Helmholtz cuts method, which occurs in a
wide theoretic and applied literature about Electromagnetism, Fluid dynamics
and Elasticity on domains of the ordinary space. We consider two classes of
bounded domains that satisfy mild boundary conditions and that become "simple"
after a finite number of disjoint cuts along properly embedded surfaces. For
the first class (Helmholtz), "simple" means that every curl-free smooth vector
field admits a potential. For the second (weakly-Helmholtz), we only require
that a potential exists for the restriction of every curl-free smooth vector
field defined on the whole initial domain. By means of classical and rather
elementary facts of 3-dimensional geometric and algebraic topology, we give an
exhaustive description of Helmholtz domains, realizing that their topology is
forced to be quite elementary (in particular, Helmholtz domains with connected
boundary are just possibly knotted handlebodies, and the complement of any
non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz
domains is a bit more advanced, and their classification appears to be a quite
difficult issue. Nevertheless, we provide several interesting characterizations
of them and, in particular, we point out that the class of links with
weakly-Helmholtz complements eventually coincides with the class of the
so-called homology boundary links, that have been widely studied in Knot
Theory.
