Classification of Prime Links in the Thickened Torus having Crossing Number 5

The goal of this paper is to tabulate all prime links in the thickened torus T x I having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus T having exactly 5 crossings. To this end. we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus T in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus T. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.