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The past few years have seen rapid progress in characterizing topologicalband structures using symmetry eigenvalue indicated methods. Recently, however,there has been increasing theoretical and experimental interest in multi-gapdependent topological phases that cannot be captured by this paradigm. Thesetopologies arise by braiding band degeneracies that reside between differentbands and carry non-Abelian charges due to the presence of either $C_2T$ or$PT$ symmetry, culminating in different invariants such as $\mathbb{Z}$-valuedEuler class. Here, we present a universal formulation for Euler phasesmotivated by their homotopy classification that is related to theSkyrmion-profile of a single unit-vector in three-level systems, and that oftwo unit-vectors in four-level systems. In addition, upon employing thestrategy of systematically building 3D models from a pair of sub-dimensionalEuler phases, we show that phase transitions between any two inequivalent Eulerphases are mediated by the presence of adjacent (in-gap) nodal rings linkedwith sub-gap nodal lines, forming trajectories corresponding to the braiding ordebraiding of nodal points. The stability of the linked adjacent nodal rings isfurthermore demonstrated to be indicated by an Euler class monopole chargematching with its $\mathbb{Z}$-valued linking numbers. We finally alsosystematically address the conversion of Euler phases into descendant Chernphases upon breaking the $C_2T$ or $PT$ symmetry. All the topological phasesdiscussed in this work are corroborated with explicit minimal lattice models.These models can themselves directly serve as an extra impetus for experimentalsearches or be employed for theoretical studies, thereby underpinning theupcoming of this nascent pursuit.

Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs