  We give an infinite family of knots such that for any given $r \geq 3$, the
family contains a knot which can be embedded on a hexagonal $r$-mosaic, but
cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing
number. This extends the square mosaic result of Ludwig, Evans, and Paat
\cite{L}. We also introduce a new tool for systematically finding all possible
flypes for the projection of any reduced, alternating prime link thus making it
easier to find all possible minimal crossing embeddings of prime, alternating
knots.
