Upper bound on the total number of knot n-mosaics

Abstract

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 x n matrix of 11 kinds of specific mosaic tiles representing a knot or a link by adjoining properly that is called suitably connected. D-n denotes the total number of all knot n-mosaics. Already known is that D-1 = 1, D-2 = 2 and D-3 = 22. In this paper we establish the lower and upper bounds on D-n 2/275 (9.6(n-2) + 1)(2) . 2((n-3)2) <= D-n <= 2/275 (9 . 6(n-2) + 1)(2) . (4.4)((n-3)2). and find the exact number of D-4 = 2594.