Small knot mosaics and partition matrices

Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an m x n matrix of mosaic tiles which are T₀ through T₁₀ depicted, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. D_m,ndenotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics. In this paper, we find a table of the precise values of D_m,n for 4 ≤ m ≤ n ≤ 6. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.