Helicity, linking, and writhe in a spherical geometry

Abstract

Linking numbers, helicity integrals, twist, and writhe all describe the topology and geometry of curves and vector fields. The topology of the space the curves and fields live in, however, can affect the behaviour of these quantities. Here we examine curves and fields living in regions exterior to a sphere or in spherical shells. The winding of two curves need not be conserved because of the topology of a spherical shell. Avoiding the presence of magnetic monopoles inside the sphere is very important if magnetic helicity is to be a conserved quantity. Considerations of parallel transport are important in determining the transfer of helicity through the foot of a magnetic flux tube when it is in motion.