Basic nets in the projective plane

TL;DR

This paper extends the theory of basic nets from the sphere to the projective plane, showing they can be generated from minimal nets via local transformations and establishing a uniqueness property related to their pull-backs.

Contribution

It proves a classification of basic nets on the projective plane analogous to the sphere case and demonstrates a unique determination of graphs on RP^2 by their pull-backs on S^3.

Findings

Basic nets on RP^2 can be generated from minimal nets using local transformations.

Graphs on RP^2 are uniquely determined by their pull-back on S^3.

The results extend Conway's enumeration approach to the projective plane.

Abstract

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).