Classification of knotted arcs in the thickenes punctured torus

In the paper we classify knotted arcs in the thickened punctured torus $T_0\times I$. Arcs are constructed by a three-step process. First we enumerate graphs with 0, 1, 2 vertices of valence 4 and two vertices of valence 1, then for each graph we enumerate all corresponding knot projections, and after that we construct the corresponding minimal diagrams. For proving that all knotted arcs are different we use the generalized Kauffman polynomial.