12 problems

A formal knot mosaic system is the square-grid representation of a knot or link using blank tiles and tiles containing portions of a knot diagram, with adjacent portions connected…

Let a knot mosaic be a suitably connected square grid of standard mosaic tiles, and let mosaic moves be the Reidemeister-type moves, planar isotopies, and size-changing operations…

Crab bucket non-destabilization conjecture. For all odd $n\geq 5$, $\beta_n$ cannot be negatively destabilized without increasing its mosaic number.

Heap's conjecture. Such space-efficient layouts produce the same set of prime knots.

Kuriya–Shehab's conjecture. The mosaic number of $6_3$ is $6$.

For a knot, let a hexagonal $r$-mosaic be a knot diagram assembled on the hexagonal $r$-mosaic arrangement, and let the interior tiles be the tiles away from the boundary. The four…

For a knot, let a hexagonal $r$-mosaic be a knot diagram assembled on the hexagonal $r$-mosaic arrangement. The interior tiles are the tiles away from the boundary. The four three-…

For each integer $k\geq 1$, let $E_{2k+1}$ denote the family of alternating knots that embed on an $M_{2k+1}$ mosaic with $|D|=2$, where $D$ is the relevant dual structure. A knot…

The mosaic number $m(L)$ of a link $L$ is the smallest integer $n$ for which $L$ is representable as a link $n$-mosaic. For the knot $2\,1\,1\,2$, also denoted $6_3$, its minimal r…

Let $K_1$ and $K_2$ be knot $n$-mosaics, and let $\lvert K_1\rangle$ and $\lvert K_2\rangle$ be the corresponding quantum knots. Let $\underset{n}{\approx}$ denote equivalence unde…

Let $k_1$ and $k_2$ be tame knots or links, and let $K_1$ and $K_2$ be arbitrary chosen mosaic representatives of $k_1$ and $k_2$, respectively. Two mosaics have the same knot mosa…

A mosaic system is the formal rewriting system of symbol strings consisting of the graded set of knot mosaics and its graded knot mosaic ambient group. Writing this system as…