Knot Mosaics for the 6_1 Knot
For many knots, the tile number and/or crossing number cannot be realized when the mosaic number is realized.
The mosaic of the 61 knot given in the table has the mosaic number and tile number realized, but the crossing number of this knot cannot be realized on a 5-mosaic. The crossing number is first realized on a 6-mosaic. Below are the relevant mosaics.
References:
Heap, A.; Knowles, D. Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6; Involve 2019, 12.
Lee, H.; Ludwig, L.; Paat, J.; Peiffer, A.Knot Mosaic Tabulation; Involve 2018, 11.
Ludwig, L.; Evans, E.An Infinite Family of Knots Whose Mosaic Number Is Realized in Non-reduce Projections; J. Knot Theory Ramif. 2013, 22.