Spherical Knot Mosaics

Abstract

In this paper we introduce the notion of a spherical knot mosaic where a knot is represented by tiling the surface of a topological 2-sphere with 11 canonical knot mosaic tiles and show this gives rise to several novel knot (and link) invariants: the spherical mosaic number, spherical tiling number, minimal spherical mosaic tiling number, spherical face number, spherical n-mosaic face number, and minimal spherical mosaic face number. We show examples where this framework offers an improvement over classical knot mosaics. Furthermore, we explore several bounds involving classical knot invariants derived from these spherical mosaic invariants.