Knot Mosaics for the 10_34 Knot

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

The mosaic of the 10₃₄ knot given in the table has the mosaic number (6) and tile number (27) realized, but the crossing number of this knot cannot be realized with only 27 non-blank tiles. The crossing number is first realized on a 6-mosaic with 32 non-blank tiles, but it can also be realized on a 7-mosaic with 29 non-blank tiles. Below are the relevant mosaics.The mosaic of the 10₁₃₉ 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 29, and this cannot be realized on anything smaller than a 7-mosaic. The crossing number is also first realized on a 7-mosaic with 29 non-blank tiles. Below are the relevant mosaics.