  We consider the phenomenon of quantum mechanical contextuality, and
specifically parity-based proofs thereof. Mermin's square and star are
representative examples. Part of the information invoked in such contextuality
proofs is the commutativity structure among the pertaining observables. We
investigate to which extent this commutativity structure alone determines the
viability of a parity-based contextuality proof. We establish a topological
criterion for this, generalizing an earlier result by Arkhipov.
