Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6

TL;DR

This paper identifies all prime knots with mosaic number 6, constructs minimal space-efficient mosaics for each, and determines their tile numbers, advancing understanding of knot representations on mosaics.

Contribution

It provides a complete classification and minimal mosaics for prime knots with mosaic number 6, a specific case not previously fully explored.

Findings

Complete list of prime knots with mosaic number 6.

Minimal space-efficient mosaics for each knot.

Tile numbers for all these knots.

Abstract

In 2008, Kauffman and Lomonaco introduce the concepts of a knot mosaic and the mosaic number of a knot or link, the smallest integer $n$ such that a knot or link can be represented on an $n$-mosaic. In arXiv:1702.06462, the authors explore space-efficient knot mosaics and the tile number of a knot or link, the smallest number of non-blank tiles necessary to depict the knot or link on a mosaic. They determine bounds for the tile number in terms of the mosaic number. In this paper, we focus specifically on prime knots with mosaic number 6. We determine a complete list of these knots, provide a minimal, space-efficient knot mosaic for each of them, and determine the tile number (or minimal mosaic tile number) of each of them.