Knotted linear force-free magnetic fields-topological aspects

The problem of computing linear force-free magnetic fields on a knotted multiply-connected domain is considered. The domain is the support of the current distribution, and the linear force-free fieldproblem reduces to finding an eigenfield of a self-adjoint curl operator. In this context, the GKN Theorem is reformulated in terms of symplectic geometry in order to characterize the self-adjoint extensions of the curl operator restricted to solenoidal vector fields. When further restricted to the isotopy invariant boundary conditions, the self-adjoint extensions are parametrized by the Lagrangian subspaces of the symplectic form on the first homology group of the boundary. This paper discusses some of the topological aspects and gives some pointers for the associated finite element discretization. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)