Flows on Simplicial Complexes

Given a graph $G$, the number of nowhere-zero ${\text{ZZ}}_q$-flows ${\varphi }_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero ${\text{ZZ}}_q$-flows to simplicial complexes $\mathrm{\Delta }$ of dimension greater than one, and prove the polynomiality of the corresponding function ${\varphi }_{\mathrm{\Delta }}(q)$ for certain $q$ and certain subclasses of simplicial complexes.