2013/12/14 by Hong, Kyungpyo, Lee, Ho, Lee, Hwa Jeong +1
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1312.4009
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 × n matrix of mosaic tiles which are T0 through T10 depicted as below, 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. Dm,n denotes 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 Dm,n for 4 ≤ m ≤ n ≤ 6 as below. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics. Dm,n · amp; n=4 · amp; n=5 · amp; n=6
m=4 · amp; 2594 · amp; 54,226 · amp; 1,144,526
m=5 · amp; · amp; 4,183,954 · amp; 331,745,962
m=6 · amp; · amp; · amp; 101,393,411,126