Step 1 — Gluing edges of a square - Genus 0
By gluing edges of a square together, one can create other shapes like a cylinder, or a sphere.
All shapes obtainable from this approach have a genus of 0, causing any graph without edge intersections in the plane to also have no edge intersections on such a shape.
Step 2 — Gluing edges of a square - Genus 1
One may glue the two opposing edges of a square to a cylinder and bend it into a torus by gluing its two openings together.
A torus has a genus of 1, some graphs that previously did not reveal a planar embedding can now be embedded.
An example of this is the Kuratowski subgraph K3,3, which is first rearranged by flipping the vertices 2 and 5 to reduce the amount of intersecting edges and then rerouting the remaining intersecting edges (2,5) and (1,6) over opposing square borders, causing them to connect again at the end of the gluing process.
Step 3 — Visualize embeddings on surfaces
By folding the square whilst twisting it 180 degrees and gluing its vertical edges together, one can obtain the infamous Möbius strip, which has also a genus of 1. Therefore we can proceed to find an embedding of K3,3 on it without any intersections. For this, we first flip the vertices 2 and 5 of the graph and reroute the remaining intersecting graph lines (3,4) and (1,6) over the vertical borders of the square.
Doing this causes them to connect again on the Möbius strip, thus yielding a planar embedding.