Mosaic number and Tile number of Corner Connection Tiles

Abstract

Lomonaco and Kauffman introduced knot mosaics in 2008 to model physical quantum states. These mosaics use a set of tiles to represent knots on $nxn$ grids. In 2023 Heap introduced a new set of tiles that can represent knots on a smaller board for small knots. Completing an exhaustive search of all knots or links, $K$, on different board sizes and types is the most common way to determine invariants for knots, such as the smallest board size needed to represent a knot, $m(K)$, and the least number of tiles needed to represent a knot, $t(K)$. In this paper, we propose a solution to an open question by providing a proof that all knots or links can be represented on corner connection mosaics using fewer tiles than traditional mosaics $t_c(K)<t(K)$, where $t_c(K)$ is the smallest number of corner connection tiles needed to represent knot \textit{K}. We also define bounds for corner connection mosaic size, $m_c(K)$, in terms of crossing number, $c(K)$, and simultaneously create a tool called the \textit{Corner Mosaic Complement} that we use to discover a relationship between traditional tiles and corner connection tiles. Finally, we construct an infinite family of links $L_n$ where the corner connection mosaic number $m_c(K)$ is known and provide a tool to analyze the efficiency of corner connection mosaic tiles.

24 pages, 14 figures