  We study a general class of easy-axis spin models on a lattice of corner
sharing even-sided polygons with all-to-all interactions within a plaquette.
The low energy description corresponds to a quantum dimer model on a dual
lattice of even coordination number with a multi dimer constraint. At an
appropriately constructed frustration-free Rokhsar-Kivelson (RK) point, the
ground state wavefunction can be exactly mapped onto a classical vertex model
on the dual lattice. When the dual lattice is bipartite, the vertex models are
bonded and are self dual under Wegner's duality, with the self dual point
corresponding to the RK point of the original multi-dimer model. We argue that
the self dual point is a critical point based on known exact solutions to some
of the vertex models. When the dual lattice is non-bipartite, the vertex model
is arrowed, and we use numerical methods to argue that there is no phase
transition as a function of the vertex weights. Motivated by these wavefunction
dualities, we construct two other distinct families of frustration-free
Hamiltonians whose ground states can be mapped onto these vertex models. Many
of these RK Hamiltonians provably host $\mathbb{Z}_2$ topologically ordered
phases.
