Classification of links of small complexity in a thickened torus

The paper contains the table of links in the thickened torus $T^2\times I$ admitting diagrams with at most four crossings. The links are constructed by a three-step process. First we enumerate all abstract regular graphs of degree 4 with at most four vertices. Then we consider all nonequivalent embeddings of these graphs into $T^2$. After that each vertex of each of the obtained graphs is replaced by a crossing of one of the two possible types, when a segment of the graph lies lower or above another segment. The words “above” and “lower” are understood in the sense of the coordinate of the corresponding point in the interval $I$. As a result, we obtain a family of diagrams of knots and links in $T^2 \times I$. We propose a number of artificial tricks that essentially reduce the enumeration and offer a rigorous proof of the completeness of the table. A generalized version of the Kauffman polynomial is used to prove that all the links are different.