Gap labeling theorems connect the spectra of Schrödinger operators to the dynamics that generate their potentials. We will focus on the main ideas in R. Johnson’s approach to gap labeling, which uses the Schwartzman group. After showing how this group can be computed in several dynamical settings, I will present results (joint with D. Damanik and J. Fillman) that describe how these dynamical invariants determine, and in some cases ensure, the opening of spectral gaps. In particular, for potentials generated by the full shift on finitely many symbols, every label predicted by the gap labeling theorem is realized at large coupling. We will conclude with a discussion of the proof and in which scenarios certain labels fail to occur.