Magnetic Writhe and Self-Organized Braiding(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

Abstract

Knot theory and the geometry of curves have important applications in astrophysics and fluid mechanics. This paper presents two. First, the writhe number, which measures the buckling and coiling of a closed curve, arises in the study of magnetic structures in the atmosphere of the sun. As these structures have endpoints at the solar surface, the definition of writhe must be modified. We present definitions for open writhe appropriate for both unconstrained open curves, and for curves with endpoints on a physical boundary. Secondly, braids occur naturally in the solar atmosphere: magnetic field lines in x-ray loops can become braided owing to motions of the endpoints at the surface. Reconnection in the atmosphere reduces the topological complexity of the magnetic field, and releases magnetic energy in the form of flares. We conjecture that the braid pattern evolves to a self-organized state with power law statistical properties.