Tile Numbers of Knot Corner Mosaics

A knot mosaic is a grid of pictorial tiles representing a tame knot or link. Recently, two groups independently introduced a new set of tiles. We call mosaics made with these new tiles corner mosaics. The (corner) tile number is the minimum number of tiles needed to represent a knot or link as a (corner) mosaic. We show that the corner tile number lies strictly between the tile number and $3$ times the tile number, resolving a question of Heap et al. We also show that the only knots and links with corner tile number $<12$ and no unlinked, unknotted components are the Hopf link $P(1,1)$, the trefoil knot $3_1$, Solomon’s knot $P(1,1,1,1)$, the connect sum of two Hopf links $P(1,1)\mathrm{\#}P(1,1)$, the cinquefoil knot $5_1$, the star of David link $P(1,1,1,1,1,1)$, the figure-eight knot $4_1$, and the three-twist knot $5_2$.