2013/03/28 by Kyungpyo Hong, Hong, Kyungpyo, Ho Lee +5
Mathematics · Computer Science · #Geometric and Algebraic Topology #semigroups and automata theory #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1303.7044
Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D1=1, D2=2, and D3=22. In this paper we establish the lower and upper bounds on Dn (2)/(275)(9 ⋅ 6n-2 + 1)2 ⋅ 2(n-3)2 ≤ Dn ≤ (2)/(275)(9 ⋅ 6n-2 + 1)2 ⋅ (4.4)(n-3)2. and find the exact number of D4 = 2594.