Noncoplanar spin textures act on double-exchange (DE) electrons as an emergent magnetic flux. For short-period spin crystals in frustrated magnets, the resulting band topology is set by the full Bloch Hamiltonian, not by the skyrmion charge alone. Here, we classify the DE band topology of field-selected noncoplanar crystals on a triangular-lattice $K\text{−}\mathrm{\Gamma }\text{−}{\mathrm{\Gamma }}^{\prime }$ magnet (with 7-, 20-, and 36-site magnetic cells), alongside canonical reference textures. Two distinct classes emerge, distinguished by the net emergent flux through the magnetic cell. Crystals with a net flux (20-site cells, skyrmion charge $Q_{\text{sk}}=\pm 4$) are Hofstadter-type Chern insulators whose global gaps persist down to weak Hund couplings of $J_H\approx 0.7t$ (where $t$ is the hopping amplitude), well below the bandwidth. In contrast, the zero-net-flux 7-site crystal realizes a Haldane-type band with $Q_{\text{sk}}=0$ yet $C=+1$. Its gap, generated solely by flux modulation, is narrow ($0.04–0.17t$) and is closed by minority-spin admixture unless $J_H$ reaches $8–38t$. The zero-net-flux 36-site crystals behave as Chern metals even in the strong-Hund limit. In both classes, the topological robustness is dictated by the strong-Hund global gap—an electronic-structure property that local texture diagnostics fail to capture. The bulk topology of the net-flux class is further confirmed by a quantized anomalous Hall plateau and a chiral edge mode. Finally, the applied field acts as a topological switch: its direction selects the topological class, while its magnitude drives the zero-net-flux branch from a Chern insulator to a Chern metal.

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• Anomalous Hall effect
• Berry curvature
• Chern insulators
• Spin texture