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 $n\times n$ 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 $D_n$ $$\frac{2}{275}(9\cdot 6^{n-2}+1{)}^2\cdot 2^{(n-3{)}^2}\le D_n\le \frac{2}{275}(9\cdot 6^{n-2}+1{)}^2\cdot (4.4{)}^{(n-3{)}^2}.$$ and find the exact number of $D_4=2594$.

6 pages, 3 figures

References in corpus (3)

• On a Lomonaco-Kauffman conjecture
• Quantizing Knots and Beyond
• Mosaic number of knots

Cited by in corpus (2)

• Mosaic number of knots
• Small knot mosaics and partition matrices