On Upper Bounds for Toroidal Mosaic Numbers

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.