Home▸Mathematics▸Mathematics Topology Simulation

🧪 Mathematics Topology Simulation

Build a T(p,q) torus knot, rotate and continuously deform it, and watch the live diagram crossing count fluctuate while the true minimal crossing number and genus — real knot invariants — stay fixed, exactly as Murasugi's theorem predicts.

Mathematics2DModerate60 FPS📱 Mobile-adapted

This simulator builds a real torus knot T(p, q) — the curve that winds p times around a torus tube and q times through its hole — as an actual 3D geometry you can rotate, and lets you continuously deform it while two numbers are tracked side by side: the crossing count of the diagram you are currently looking at, and the knot's true minimal crossing number and genus, computed from Murasugi's exact formulas for torus knots. Rotating or wobbling the knot changes the live diagram count constantly, while the invariants never move — a direct, hands-on demonstration of what a knot invariant actually is and why topologists need them to tell knots apart.

⚙ Under the hood

This simulation provides an interactive introduction to topology, exploring concepts like connectivity, holes, and surfaces. Users can manipulate shapes and observe how topological changes affect their properties.

TopologyMathVisualization

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install