Classification of Low Complexity Knots in the Thickened Torus

We compose the table of knots in the thickened torus $ T\times I$ which have diagrams with $\leq 4$ crossing points. The knots are constructed by a three-step process. First we enumerate regular graphs of degree 4, then for each graph we enumerate all corresponding knot projection, and after that we construct the corresponding minimal diagrams. Several known and new tricks made possible to keep the process within reasonable limits and offer a rigorous theoretical proof of the completeness of the table. For proving that all knots are different we use a generalized version of the Kauffman polynomial.