Multifold band degeneracies and linking of nodal manifolds

Topological degeneracies of energy bands in crystalline matter are conventionally characterized by invariants computed on an enclosing sphere over which the spectrum remains gapped. Crucially, for higher-order degeneracies, the enclosing-sphere paradigm faces a fundamental obstruction: two (n − 1)-fold degeneracy loci emanating from the n-fold node necessarily pierce every choice of enclosing sphere, leaving no uniform spectral gap and thus no standard homotopy classification. However, we can elevate this obstruction into the diagnostic itself. Namely, on the two nodal manifolds where the loci cross the sphere, complementary spectral gaps are restored, admitting conventional band invariants (Chern numbers, Stiefel-Whitney classes, and winding numbers). This observation establishes a general two-way correspondence: (1) the multifold node is topologically protected whenever the associated nodal manifolds are robustly linked, and (2) invariants on cycles of one manifold encode their linking numbers with cycles of the other. Carrying out this program, we can recast multifold band topology as the topology of linked nodal manifolds.