Articles 51376 and 89053 describe ways to construct models involving quantum gauge fields when space or spacetime is treated as a lattice. In those models, the gauge field is described as a collection of G-valued link variables, one for each directed link in the lattice, where G is the gauged group. The collection of link variables used in those constructions is overcomplete: the same model (on the same lattice) can be described using only a subset of the link variables. This article explains how to choose a subset that is just barely complete. This is a type of gauge fixing. Important special cases of this type of gauge fixing include the temporal gauge and the axial gauge (after extending them slightly so that only a minimal complete set of link variables remains). This type of gauge fixing can be useful for understanding a model's general properties. Gauge fixing is also a prerequisite for defining small-coupling expansions. This article also shows that even after complete gauge fixing, some zero modes may remain.
