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.