Knot Mosaics for the 12a_169 Knot

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

The mosaic of the 12a169 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 31, and this cannot be realized on anything smaller than a 7-mosaic. Below are the relevant mosaics.
12a169 with mosaic number realized
12a169 with tile number realized

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References:

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.; Baldwin, D.; Canning, J.; Vinal, G. Knot Mosaics For Prime Knots with Crossing Number 10 or Less; in preparation.