Cohomology of knotted semimetals in three dimensions

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 1-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 neighbourhood surrounding a Weyl submanifold via the Mayer-Vietoris sequence, under the asumption that a particular homology group associated to the Brillouin zone and the nodal lines is torsion-free.