An infinite family of knots whose hexagonal mosaic number is only realized in non-reduced projections

We give an infinite family of knots such that for any given $r\ge 3$, 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 rectangular mosaic result of Ludwig, Evans, and Paat. We also introduce a new tool for systematically finding all possible flypes for the diagram of any link thus making it easier to find all possible minimal crossing embeddings of prime, alternating knots.