Cohomology of knotted semimetals in three dimensions
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(Embargo ends 28 February 2027)
Date
2026
Authors
Celeste, J.
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International Journal of Geometric Methods in Modern Physics, 2026; 1-16
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Joshua Celeste
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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 T³. These nodal lines have two charges, the quantized Berry phase and the Z₂-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 Z₂ 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.
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