Spherical Knot Mosaics

TL;DR

The paper extends knot mosaics to spherical surfaces by tiling the faces of an $n \times n \times n$ cube to obtain spherical knot mosaics on $S^2$, and defines a suite of new invariants including $sm(K)$, $st(K)$, $st_M(K)$, $sf(K)$, $sf_n(K)$, and $sf_M(K)$. It demonstrates that these invariants can capture knot structure beyond classical mosaics, provides planar representations for the first 35 knots, and establishes bounds linking them to mosaic and crossing numbers. The results offer sharper distinctions in knot mosaic theory and lay groundwork for further refinement of spherical mosaic invariants. The work suggests new directions for understanding how spatial embedding on a spherical surface influences knot representations and associated combinatorial limits.

Abstract

In this paper we introduce the notion of a spherical knot mosaic where a knot is represented by tiling the surface of an n by n by n 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 tile number, spherical face number, spherical n-mosaic face number, and minimal spherical mosaic face number. We show examples where this framework is an improvement over classical knot mosaics. Furthermore, we explore several bounds involving other classical knot invariants that these spherical mosaic invariants gives rise to.

Figures (6)

• Figure 1: The 11 Mosaic Tiles
• Figure 2: Knot mosaics of the unknot (left), $4_1$ (center), and $7_4$ (right) Lee_2018
• Figure 3: An example of the unknot as a spherical 1-mosaic using three tiles, $T_2$, $T_3$, and $T_4$.
• Figure 4: Planar representation of the cube with identified edges color-coded.
• Figure 5: Three examples of a planar representation of the unknot.

Theorems & Definitions (26)

• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Remark 4.3
• Example 4.4
• Proposition 4.5
• proof
• Proposition 5.1
• proof