TOD

Abstract

We consider the helicity of a vector field, which calculates the average linking number of the field’s flowlines. Helicity is invariant under certain diffeomorphisms of its domain – we seek to understand which ones.

Extending to differential (k+1)-forms on domains R^{2k+1}, we express helicity as a cohomology class. This topological approach allows us to find a general formula for how much helicity changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus. This approach also leads us to define submanifold helicities: differential (k+1)-forms on n-dimensional subdomains of R^m.