Topology-based inequalities and inverse problems for near force-free magnetic fields

We review a conjecture characterizing the knotting of current paths arising as solutions to an inverse problem involving near force-free magnetic fields. Results about the nonexistence of solutions involving force-free fields supported in a finite domain are then considered, as are explicit constructions of force-free solutions in unbounded domains. This shows why truncating solutions defined on unbounded domains has proven ineffective in the literature, and why the solution to the inverse problem involves a "near force-free magnetic field". Solutions are then characterized by inequalities involving the current distribution's mean asymptotic linking and crossing numbers. The inequalities are related to the invariants involved in the conjecture of Crager and Kotiuga.