Cohomology of knotted semimetals in three dimensions

In this paper, we extend the topological classification scheme of Weyl semimetals via cohomology and the Mayer–Vietoris sequence to account for nodal line semimetals with space-time inversion symmetry. These are semimetals where bands meet in one-dimensional submanifolds called nodal lines. These are generally a union of knots in [Formula: see text]. These nodal lines have two charges, the quantized Berry phase and the [Formula: see text]-monopole charge, the second related to linking numbers of nodal knots between bands. We provide a manifestly topological proof of the Weyl charge cancellation condition for the [Formula: see text] monopole charge, which is known to be the second Stiefel–Whitney class of a tubular neighborhood surrounding a Weyl submanifold via the Mayer–Vietoris sequence, under the assumption that a particular homology group associated to the Brillouin zone and the nodal lines is torsion-free.