Read the explanation on the left, explore the visualization on the right.
Module 1
Step 1 — What is topology?
Topology is a branch of mathematics that studies properties of shapes that stay the same under continuous deformation. Distances and angles do not matter.
Allowed: stretching, twisting, bending, crumpling Not allowed: tearing, gluing, creating holes, closing holes, passing through itself
Example: Shapes in the right panel are topologically the same. A sphere and a cube are hence equivalent.
Those "shapes" will be the surfaces we study the embedding of graphs on.
Step 2 — Orientable Surfaces
Surfaces can be classified into orientable and nonorientable surfaces.
Orientable surfaces admit a globally consistent orientation. This means you can move along the surface without suddenly finding that "top" has become "bottom".
Example: A cylinder is orientable
Move an arrow pointing "top" around the cylinder once, and it still points "top" when it returns. The orientation remains consistent.
turns = 0.000
Step 3 — Non-Orientable Surfaces
A Möbius strip is non-orientable.
Move the arrow around the strip once, and when it returns, it points in the opposite direction. The orientation has flipped!
turns = 0.000
Step 4 — Homeomorphism + genus
The shown graph has an intersection when embedded on the sphere. Add a handle to the sphere and route one of the intersecting edges through it. Now the graph can be embedded without intersections!
The number of handles a surface has is called its genus.
Two surfaces are homeomorphic if one can be continuously deformed into the other without tearing, gluing, or creating/removing holes. Therefore, surfaces with the same genus are homeomorphic.
Genus and Embedding Demo
Press the "Show Handle" button to resolve the intersecting edges by adding a new handle.
Remark: For non-orientable surfaces, crosscaps are used instead of handles (a Möbius-strip-like attachment). In that case, homeomorphism is classified by the number of crosscaps instead.