Quantum knot mosaics and the growth constant

Abstract

Lomonaco and Kauffman introduced a knot mosaic system to give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot n-mosaic is an nxn array of 11 mosaic tiles representing a knot or a link diagram by adjoining properly that is called suitably connected. The total number of knot n-mosaics is denoted by D-n which is known to grow in a quadratic exponential rate. In this paper, we show the existence of the knot mosaic constant delta = lim(n ->infinity) Dn1/n(2) and prove that 4 <= delta <= 5+root 13/2 (approximate to 4.303) (C) 2016 Elsevier B.V. All rights reserved.