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 × 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 [Formula: see text] and find the exact number of D 4 = 2594.