Basic nets in the projective plane

The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway’s approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $R{P}^3$ using basic nets on $R{P}^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets (the nets $(2\times n{)}^{\ast }$, $n\ge 3$, in Conway’s notation) by two local transformations. We prove a similar result for basic nets in $R{P}^2$. We prove also that a graph on $R{P}^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).