# Tile Number and Space-Efficient Knot Mosaics

## TL;DR

This paper introduces the concept of space-efficient knot mosaics, establishing bounds for the minimal number of tiles needed to depict knots, and provides specific examples of such mosaics for various knots.

## Contribution

It defines the tile number for knots, derives bounds based on mosaic number, and constructs space-efficient mosaics for specific knots.

## Key findings

- Bounds for tile number in terms of mosaic number are established.
- Exact tile numbers are determined for several knots.
- Space-efficient knot mosaics are constructed for multiple examples.

## 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 $5m-8 \leq t \leq m^2-4$ if $m$ is even and $5m-8 \leq t \leq m^2-8$ if $m$ is odd. We also determine the tile number of several knots and provide space-efficient knot mosaics for each of them.