  Nodal lines inside the momentum space of three-dimensional crystalline solids
are topologically stabilized by a $\pi$-flux of Berry phase. Nodal-line rings
in $\mathcal{PT}$-symmetric systems with negligible spin-orbit coupling (here
described as "nodal class AI") can carry an additional "monopole charge", which
further enhances their stability. Here, we relate two recent theoretical
advancements in the study of band topology in nodal class AI. On the one hand,
cohomology classes of real vector bundles were used to relate the monopole
charge of nodal-line rings to their linking with nodal lines formed among the
occupied and among the unoccupied bands. On the other hand, homotopy studies
revealed that the generalization of the Berry-phase quantization to the case of
multiple band gaps defines a non-Abelian topological charge, which governs the
possible deformations of the nodal lines. The present work has three aims.
First, we present how to efficiently wield the recently discovered non-Abelian
topological charge. Second, we apply these methods to present an independent
proof of the relation between the monopole charge and the linking structure,
including all the fragile-topology exceptions. Finally, we show that the
monopole charge flips sign when braided along a path with a non-trivial Berry
phase. This facilitates a new type non-Abelian "braiding" of nodal-line rings
inside the momentum space, that has not been previously reported. The work
begins with a brief review of $\mathcal{PT}$-symmetric band topology, and the
geometric arguments employed in our theoretical analysis are supplemented in
the appendices with formal mathematical derivations.
