J. Cantarella et al., The Biot-Savart operator for application to knot theory, fluid dynamics, and plasma physics, J MATH PHYS, 42(2), 2001, pp. 876-905
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.