Topology Explorer

Explore Euler characteristics, surfaces, Möbius strips, and knots.

Topology & Manifold Invariants Handbook

Mathematical Principles & Theorems

Investigates topological invariants preserved under homeomorphisms (continuous bijective maps with continuous inverses). Euler Characteristic \(\chi = V - E + F = 2 - 2g\) classifies closed orientable surfaces by genus \(g\) (Sphere \(g=0, \chi=2\); Torus \(g=1, \chi=0\); Double Torus \(g=2, \chi=-2\)). Non-orientable surfaces: Möbius Strip (one-sided, single continuous boundary curve, \(\chi=0\)) and Klein Bottle (closed non-orientable surface with no boundary, \(\chi=0\)).

Operating Instructions

Select topological manifold: Möbius Strip, Torus, Klein Bottle, or Sphere.
Configure rotation speed, mesh resolution, and twist count using numeric inputs.
Click and drag on the 3D WebGL stage to inspect surface continuity and parametric coordinates.
Review topological invariant metrics (Euler characteristic \(\chi\), genus \(g\), orientability, boundary components).

Controls

Vertices V

Edges E

Faces F

Topological Surfaces & Manifolds. Browse the library of standard surfaces on the right.

Rotation speed

Half-twists

Select a knot to visualize:

χ = V−E+F —

Surface —

Genus g —

Betti numbers —

Euler's Formula for Polyhedra

For any convex polyhedron: χ = V − E + F = 2
For a surface of genus g (g handles): χ = 2 − 2g
Sphere: χ=2 (g=0) | Torus: χ=0 (g=1) | Double-torus: χ=−2 (g=2)

Shape	V	E	F	χ
Tetrahedron	4	6	4	2
Cube	8	12	6	2
Octahedron	6	12	8	2
Torus	1	2	1	0
Klein Bottle	1	2	1	0

Sides 1

Edges 1

Orientable? No

χ 0