Knot Mosaics for the 10<sub>61</sub> Knot

Knot Mosaics for the 10₆₁ Knot

For many knots, the tile number and/or crossing number cannot be realized when the mosaic number is realized. The mosaic of the 10₆₁ 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, but at least 31 non-blank tiles are necessary to achieve it. Below are the relevant mosaics.

10₆₁ with mosaic number realized.			10₆₁ with tile number realized.		10₆₁ with crossing number realized.

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References:

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.