On upper bounds for toroidal mosaic numbers

In this paper, we construct mosaic representations of knots on the torus, rather than in the plane. This consists of a particular choice of the ambient group $mathbb{A }$ , 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 (called waste) 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 torus $2$ -mosaics.