Classification of Links of Small Complexity in the Thickened Torus

Abstract

We present a complete table of links in the thickened torus T² × I of complexity at most 4. The table is constructed by the following method. First, we enumerate all abstract quadrivalent graphs with at most four vertices. Then we consider all inequivalent embeddings of these graphs in the torus. After that each vertex of each of the obtained graphs is replaced by a crossing of one of two possible types specifying the over-strand and the under-strand. The words “over” and “under” are understood in the sense of the coordinate of the corresponding point in the interval I. As a result, we obtain a family of link diagrams in the torus. We propose a number of artificial tricks that essentially reduce the enumeration and help to prove rigorously the completeness of the table. We use a generalized version of the Kauffman polynomial to prove that the obtained links are pairwise inequivalent.