Magnetic topology in coupled binaries

We consider topological configurations of the magnetically coupled spinning stellar binaries. We discuss conditions when the stellar spins and the orbital motion `compensate' each other, leading to periodic untwisting of the magnetosphere; such untwisting can be global and/or local. We describe the topology of the relevant space as $SO(3)=\mathbb{R}P^3$ or $\mathbb{F}=STS^2$ and find conditions for `unwinding' configurations in terms of magnetic moments, spins and orbital momentum. These conditions become ambiguous near topological bifurcation points as they depend on the details of the magnetic field dynamics.