It is well known that any virtual link diagram can be represented as a link diagram without virtual crossings on a closed orientable surface in the \(3\)-sphere. In fact, there is a canonical way to construct such a link diagram by N. Kamada and S. Kamada [J. Knot Theory Ramifications 9, No. 1, 93–106 (2000; Zbl 0997.57018)]. Thus the supporting genus of a virtual link diagram is defined to be the minimal genus among genera of surfaces that can contain such link diagram without virtual crossings of the virtual link diagram. Also, the supporting genus of a virtual link is defined to be the minimal among supporting genera of its diagrams. This notion is naturally generalized to a projected virtual link diagram, which is a virtual link diagram ignoring the information at all ordinary crossings. As an equivalence class of a projected virtual link diagram under some set of local moves called projected virtual Reidemeister moves, a projected virtual link is defined.
The author defines the notion of a reduced diagram for a projected virtual link diagram, and shows that any projected virtual link has a reduced diagram and that any two reduced diagrams are related to each other by a finite sequence of the projected Reidemeister move III. Furthermore, it is proved that any reduced diagram of a projected virtual link realizes the supporting genus of the link. By using this, the author detects the non-triviality of Kishino’s virtual knot.