Spherical Knot Mosaics

TL;DR

This paper introduces spherical knot mosaics, a novel way to represent knots on a sphere using tilings, leading to new invariants and potential improvements over classical knot mosaics.

Contribution

The paper defines spherical knot mosaics, introduces related invariants, and demonstrates their advantages and bounds compared to traditional knot mosaic methods.

Findings

New spherical mosaic invariants for knots and links

Examples showing improvements over classical mosaics

Bounds relating spherical mosaic invariants to classical knot invariants

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.