The band structure of crystals has degenerate energies in two or more bands and peculiarities in the shape of the Bloch state manifold, which can affect material and transport properties. Ultracold atoms in optical lattices have been used to characterize such points only indirectly, for example by detection of Abel-Berry phases, and only at singularities with linear dispersion (Dirac points). Here we investigate singularities in the band structure through non-Abelian transformations generated by transport directly through the singularities. We prepare atoms in one Bloch band and accelerate them along quasi-momentum trajectories that enter, reorient, and exit the singularity at the linear and quadratic contact points of the honeycomb lattice. Measurement of the band population after transport identifies the winding numbers of these singularities as 1 and 2, respectively. Our work opens up the study of second-order band contact points in cryogenic atomic quantum simulators and also provides new methods for probing singularities in other band shapes.

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