On knotoids and their relation with knots of geometric degree 1 in the thickened torus

Knotoid theory was created by V. Turaev in 2011. As classical knots, knotoids are represented by their diagrams on the 2-sphere. A knotoid diagram is a curve with self-intersections eqipped by over/undercrossing data. We define the equivalence relation on the set of knotoid diagrams and define some invariants of knotoids, as the Kauffman polinomial. The thickened torus is a direct product of the 2-torus and the interval I. A knot in the thickened torus is a simple closed curve. We define an equivalence relation on the set of knots in the thickened torus. We define the notion of knots of geometric degree 1 in the thickened torus and we construct the lifting map. This map gives a correspondence between knots of geometric degree 1 in the thickened torus and knotoids on the 2-sphere. We construct the switch-operation for knotoids and define prime knotoids. We prove the following theorem: the lifting map is injective for prime knotoids of complexity at least 2.