Quantum knots and the number of knot mosaics

Abstract

Lomonaco and Kauffman developed a knot mosaic system to introduce a precise and workable definition of a 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 (T-0 through T-10 depicted in the introduction) representing a knot or a link by adjoining properly that is called suitably connected. D(m, n) is the total number of all knot (m, n)-mosaics. This value indicates the dimension of the Hilbert space of these quantum knot system. D-(m,D- n) is already found for m, n <= 6 by the authors. In this paper, we construct an algorithm producing the precise value of D-(m,D- n) for m, n >= 2 that uses recurrence relations of state matrices that turn out to be remarkably efficient to count knot mosaics. D(m,n) = 2 parallel to (Xm-2 + Om-2)(n-2) parallel to where 2(m-2) x 2(m-2) matrices Xm-2 and Om-2 are defined by Xk+1 = [X-k O-k] [O-k X-k] and Ok+1 = [O-k X-k] [X-k 4 O-k] for k = 0, 1, .... , m - 3, with 1 x 1 matrices X-0 = [1] and O-0 = [1]. Here parallel to N parallel to denotes the sum of all entries of a matrix N. For n = 2, (Xm-2 + Om-2)(0) means the identity matrix of size 2(m-2) x 2(m-2).