Tile Number and Space-Efficient Knot Mosaics

Abstract

In this paper we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in terms of the mosaic number of the knot. In particular, if $t$ is the tile number of a prime knot with mosaic number $m$, then $5m-8\le t\le m^2-4$ if $m$ is even and $5m-8\le t\le m^2-8$ if $m$ is odd. We also determine the tile number of several knots and provide space-efficient knot mosaics for each of them.

The original version of this article was split into two articles during refereeing

References in corpus (1)

• Knot mosaic tabulation

Cited by in corpus (3)

• Space-Efficient Prime Knot 7-Mosaics
• Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6
• Tabulating Knot Mosaics: Crossing Number 10 or Less