Links, Knots, and Braids

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 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 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.


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Last update 01/23/2009
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