This paper rethinks how to tell when several energy bands meet at a single point in a material or a model. The authors study generic n-fold band degeneracies — points where n energy levels coincide — and show that these crossings are topological when certain lower-dimensional “nodal” manifolds are linked around the crossing. In plain terms: instead of looking only at the energy gap on a surrounding sphere, one must look at how other band-touching sets thread that sphere.

The work focuses on degeneracies protected by Altland–Zirnbauer (AZ) symmetries. These are the ten symmetry types commonly used in condensed-matter physics to classify systems with time-reversal, particle-hole, or chiral symmetries. The authors treat symmetry acting locally in momentum space, and they study minimal models that realize an n-fold node. A key technical finding is that the codimension of a generic n-fold node grows roughly like n^2. Codimension means how many independent parameters (for example momentum directions or external tuning knobs) you must vary to reach the degeneracy. The quadratic growth implies that multifold nodes typically sit in spaces that combine physical momenta with extra tuning parameters or synthetic dimensions.

This large codimension creates a problem for the standard way of assigning topological invariants. The usual method is to enclose a node with a sphere on which the bands are gapped, and then compute an invariant on that sphere. The authors show this is not possible for generic n-fold nodes because two (n−1)-fold degeneracy sets pass through every such sphere. Those sets pierce any enclosing sphere and destroy the uniform spectral gap there. In other words, the standard “gap on a surrounding sphere” trick fails.

Rather than treating that failure as an obstacle, the authors turn it into a tool. On the curves or surfaces where those two (n−1)-fold sets meet the sphere, complementary gaps reappear: one gap closes where the other stays open, and vice versa. This lets them compute conventional band invariants on these nodal manifolds. They prove a two-way correspondence: if the nodal manifolds are robustly linked on the enclosing sphere, the central n-fold node is topologically protected; and the usual band invariants defined on cycles of one nodal manifold record the linking numbers with cycles of the other.