For many knots, the tile number and/or crossing number cannot be realized when the mosaic number is realized.

The mosaic of the 10_{21} knot given in the table has the mosaic number realized, which is 6, but the tile number and crossing number of this knot cannot be realized on a 6-mosaic. On a 6-mosaic, the fewest number of non-blank tiles needed to create this knot is 32. However, the tile number of this knot is 27, and this cannot be realized on anything smaller than a 7-mosaic. The crossing number is also first realized on a 7-mosaic, but at least 29 non-blank tiles are necessary to achieve it. Below are the relevant mosaics.

Heap, A.; Knowles, D.Tile Number and Space-Efficient Knot Mosaics; J. Knot Theory Ramif. 2018, 27. Heap, A.; Knowles, D.Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6; Involve 2019, 12. Heap, A.; LaCourt, N.Space-Efficient Prime Knot 7-Mosaics; Symmetry 2020, 12. Heap, A.; Baldwin, D.; Canning, J.; Vinal, G.Knot Mosaics For Prime Knots with Crossing Number 10 or Less; in preparation.