  We compose the table of knots in the thickened torus T x I having diagrams
with at most 4 crossings. The knots are constructed by the three-step process.
First we list regular graphs of degree 4 with at most 4 vertices, then for each
graph we enumerate all corresponding knot projections, and after that we
construct the corresponding minimal diagrams. Several known and new tricks made
it 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.
