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 ex tent to which the curve wraps and coils around itself; it has proved its im portance for molecular biologists in the study of knotted DNA and of the en zymes which affect it. The helicity of a vector field defined on a domain i n 3-space is the standard measure of the extent to which the field lines wr ap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot-Savart operator associates with each current d istribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields wh ich are tangent to the boundary and which minimize energy for given helicit y provide models for stable force-free magnetic fields in space and laborat ory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot-Savart operator. Information ab out these fields can be converted into bounds on the writhing number of a g iven piece of DNA. The purpose of this paper is to reveal new properties of the Biot-Savart operator which are useful in these applications. (C) 2001 American Institute of Physics.