Basic nets in the projective plane

S. Yu. Orevkov

Published 2013-07-28, updated 2014-09-16Version 3

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 $RP^3$ using basic nets on $RP^2$. 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 $(2\times n)^*$, $n\ge3$, in Conway's notation) by two local transformations. We prove a similar result for basic nets in $RP^2$. We prove also that a graph on $RP^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).