Space-efficient knot mosaics for prime knots with mosaic number 6

Abstract

In 2008, Kauffman and Lomonaco introduced the concepts of a knot mosaic and the mosaic number of a knot or link $K$, the smallest integer $n$ such that $K$ can be represented on an $n$-mosaic. In 2018, the authors of this paper introduced and explored space-efficient knot mosaics and the tile number of $K$, the smallest number of nonblank tiles necessary to depict $K$ on a knot 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.