On Upper Bounds for Toroidal Mosaic Numbers

TL;DR

This paper explores the construction of knot mosaics on the torus, analyzing how different configurations affect mosaic numbers and providing tools and catalogs for such representations.

Contribution

It introduces new methods for constructing toroidal knot mosaics, examines how edge identification order impacts the resulting knots, and provides a comprehensive catalog of 2x2 torus mosaics.

Findings

Mosaic numbers can decrease through specific projections on the torus.

Order of edge identification affects the resulting knot type.

Catalog of all 2x2 torus mosaics is provided.

Abstract

In this paper, we work to construct mosaic representations of knots on the torus, rather than in the plane. This consists of a particular choice of the ambient group, as well as different definitions of contiguous and suitably connected. We present conditions under which mosaic numbers might decrease by this projection, and present a tool to measure this reduction. We show that the order of edge identification in construction of the torus sometimes yields different resultant knots from a given mosaic when reversed. Additionally, in the Appendix we give the catalog of all 2 by 2 torus mosaics.