Abstract
Topological insulator-based methods underpin the topological classification of gapped bands, including those surrounding semimetallic nodal defects. However, multiple bands with gap-closing points can also possess nontrivial topology. We construct a general wave-function-based "punctured-Chern" invariant to capture such topology. To show its general applicability, we analyze two systems with disparate gapless topology: (1)a recent two-dimensional fragile topological model to capture the various band-topological transitions and (2)a three-dimensional model with a triple-point nodal defect to characterize its semimetallic topology with half integers that govern physical observables such as anomalous transport. This invariant also gives the classification for Nexus triple points (Z×Z) with certain symmetry restrictions, which is reconfirmed by abstract algebra.
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