  Lomonaco and Kauffman introduced knot mosaics in 2008 to model physical
quantum states. These mosaics use a set of tiles to represent knots on $n x n$
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.
