Cohomology of knotted semimetals in three dimensions

Abstract

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 generally in 1-dimensional submanifolds, which can generally be knots in $\T^3$. These nodal loops have two charges, the quantized Berry phase and the $\Z_2$-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 $\Z_2$ 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.