Mosaic number of knots

Abstract

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent 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. The mosaic number $m(K)$ of a knot $K$ is the smallest integer $n$ for which $K$ is representable as a knot $n$-mosaic. In this paper we establish an upper bound on the mosaic number of a knot or a link $K$ in terms of the crossing number $c(K)$. Let $K$ be a nontrivial knot or a non-split link except the Hopf link. Then $m(K)\le c(K)+1$. Moreover if $K$ is prime and non-alternating except $6_3^3$ link, then $m(K)\le c(K)-1$.

7 pages, 8 figures

References in corpus (3)

• On a Lomonaco-Kauffman conjecture
• Quantizing Knots and Beyond
• Upper bound on the total number of knot $n$-mosaics

Cited by in corpus (2)

• Upper bound on the total number of knot $n$-mosaics
• Small knot mosaics and partition matrices