In both hydrodynamics and electromagnetism, there is interest in the possibility
that divergence-free 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
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 N-1 form on a space of N dimensions. If the N-1 form is closed (implying that the direction field is divergence free) then the closed integrals of such N-1 forms are deformation invariants (hence represent topological properties) of all evolutionary processes that can be represented by a single parameter semi-group.
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 3-form vol =rho(Jx,Jy,Jz) *( dJx^dJy^dJz ). Then
contract the volume 3-form with the vector field. The result is a 2-form, a
Current, that depends upon the 3-functions that make up J. It is possible to
find an infinite number of integrating factors, or multipliers, that will cause
this 2-form to be closed.
Cartan developed a trick of prolongation where each differential use to
construct a given closed p-form is expanded in terms of a 1-parameter 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.