The Biot–Savart operator for application to knot theory, fluid dynamics, and plasma physics

The writhing number of a curve in 3-space is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot–Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot–Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot–Savart operator which are useful in these applications.