9 problems

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.

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-…

Let $K$ be a prime knot and let a space-efficient 7-mosaic of $K$ be a 7-mosaic whose tile number is minimal among mosaics of $K$. Suppose that every column or every row of the mos…

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…

For a positive integer $n$, an $n$-mosaic is an $n\times n$ mosaic-tile board representing a link or knot, and the mosaic number of a knot is the smallest natural number $m$ for wh…

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…