Knot Mosaics for the 10139 Knot

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

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.

10139 with mosaic number realized
10139 with tile number and crossing number realized

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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.

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