  We study a short-range resonating valence bond (RVB) wave function with
diagonal links on the square lattice that permits sign-problem free wave
function Monte-Carlo studies. Special attention is given to entanglement
properties, in particular, the study of minimum entropy states (MES) according
to the method of Zhang et. al. [Physical Review B {\bf 85}, 235151 (2012)]. We
provide evidence that the MES associated with the RVB wave functions can be
lifted from an associated quantum dimer picture of these wave functions, where
MES states are certain linear combinations of eigenstates of a 't Hooft
"magnetic loop"-type operator. From this identification, we calculate a value
consistent with $\ln(2)$ for the topological entanglement entropy directly for
the RVB states via wave function Monte-Carlo. This corroborates the
$\mathbb{Z}_{2}$ nature of the RVB states. We furthermore define and elaborate
on the concept of a "pre-Kasteleyn" orientation that may be useful for the
study of lattices with non-planar topology in general.
