Cuts for the magnetic scalar potential in knotted geometries and force-free magnetic fields

The geometry of current-carrying conductors giving rise to near force-free magnetic-field configurations, where the current flow is almost parallel to the magnetic-field vector, is examined. Such configurations are highly desirable for applications where the mechanical strength of the conducting material presents a problem. The research presented here argues that for a given weighted power dissipation |J|²|B|², the solution that minimizes the maximum Lorentz force at a point involves knotted current paths; a family of torus knots is proposed as a near optimal solution. We formulated a conjecture relating the force-free problem to the Alexander and Thurston norms defined on the first cohomology group of the space exterior to the knotted current paths. The conjecture states that these two norms coincide for complements of force-free current distributions