In both hydrodynamics and electromagnetism, there is interest in the possibility
that divergencefree direction field lines (usually frozen in lines of vorticity
or magnetic field) are linked or knotted. The evolution or the creation of such
a topological state is more or less a controversial unsolved problem in terms of
classical methods.
Cartan's methods of exterior differential forms can be used to construct the
integrals of Links, Braids, Torsion_Helicity and Spin, in a coordinate free
manner, and to demonstrate the topological properties of such link and braid
integrals as deformation invariants. The concept of a direction field in the Cartan calculus is represented by a N1 form on a space of N dimensions. If the N1 form is closed (implying that the direction field is divergence free) then the closed integrals of such N1 forms are deformation invariants (hence represent topological properties) of all evolutionary processes that can be represented by a single parameter semigroup.
In 3 dimensional space, closed two forms have been used to describe links.
In 4 dimensional space, closed three forms can be used to describe braids.
For example, consider a 3 dimensional domain, and an arbitrary vector field, J.
Construct the volume element 3form vol =rho(Jx,Jy,Jz) *( dJx^dJy^dJz ). Then
contract the volume 3form with the vector field. The result is a 2form, a
Current, that depends upon the 3functions that make up J. It is possible to
find an infinite number of integrating factors, or multipliers, that will cause
this 2form to be closed.
Cartan developed a trick of prolongation where each differential use to
construct a given closed pform is expanded in terms of a 1parameter group of
motions. A different parameter is used for each expansion. For example, a two
form over dx and dy then becomes a two form over d(t1) and d(t2). Integrations
over closed sets reduces to integration along closed loops. If the Loops are linked, the classic Gauss integral becomes a Link integral.
