Topology & Knot Theory Visualiser

Explore surfaces, knots, and the topology of spaces through interactive 3D visualisations

3D Surface View

Click and drag to rotate. Scroll to zoom.

Select Surface

Properties

Parameters

Description

Interactive Simplicial Complex

Click to add vertices. Shift+click two vertices to add an edge. Ctrl+click inside a triangle of edges to add a face. Right-click a vertex to delete it.

Euler Characteristic

V − E + F = χ

Preset Examples

Counts

About

Euler's formula: V - E + F = χ
For any convex polyhedron, χ = 2.
For a torus triangulation, χ = 0.
The Euler characteristic is a topological invariant.

3D Knot View

Click and drag to rotate. Scroll to zoom.

Select Knot

Torus Knot Parameters

Knot Invariants

Description

Fundamental Polygon

Select Surface

Edge Identification

Animation

1.0

Properties

About

Fundamental polygons represent closed surfaces by identifying edges of a polygon.
Arrows show which edges are glued together and in what orientation.
Same letter = same edge. Opposite arrows = reverse gluing.