Tile Numbers of Knot Corner Mosaics
Ezra Aylaian
TL;DR
The paper analyzes corner mosaics as an alternative knot mosaic framework and defines the corner tile number $t_C(L)$ alongside the edge tile number $t(L)$. It proves two key bounds: $t_C(L) + \mathrm{caps}(L) \le t(L)$ with $\mathrm{caps}(L) \ge 4$ for links with no unlinked, unknotted components, and $t(L) \le 3\,t_C(L) - 2$, placing $t_C(L)$ strictly between $t(L)$ and $3\,t_C(L)$. A tile-number preserving bijection between edge mosaics and checkerboard corner mosaics underpins these inequalities, while caps and connection points quantify the efficiency gap between the two tile systems. The authors perform a computer-assisted classification of links with small corner tile numbers, showing that for $t_C(L) < 12$ and no unlinked components, exactly eight knots/links occur, with explicit representatives listed. Overall, the work demonstrates that corner mosaics can encode links more efficiently than edge mosaics in many cases and provides a concrete finite catalogue of minimal corner tile representatives for small $t_C(L)$.
Abstract
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 $P(1,1) \# P(1,1)$, 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 $5_2$.