  We study the quench dynamics of the $q$ Potts model on different
bi/tri-dimensional lattice topologies. In particular we are interested in
instantaneous quenches from $T_i \rightarrow \infty$ to $T \leq T_s$, where
$T_s$ is the (pseudo)-spinodal temperature. The goal is to explain why, in the
large-$q$ limit, the low-temperature dynamics freezes on some lattices while,
on others, the equilibrium configuration is easily reached. The cubic ($3d$)
and the triangular ($2d$) lattices are analysed in detail. We show that the
dynamics blocks when lattices have acyclic \textit{unitary structures} while
the system goes to the equilibrium when these are cyclic, no matter the
coordination number ($z$) of the particular considered lattice.
