Knots

Proving a knot can’t be untied

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A mathematical knot is a closed loop in space.
• Tricolourability is an invariant: it never changes under wiggling.
• The trefoil is tricolourable and the unknot isn’t, so the trefoil can’t be untied.

Steps

1. Mathematical knots
2. Tricolouring
3. So what?
4. Counting colourings
5. The figure-eight knot
6. What did we learn?