In the 1930s, Kurt Reidemeister proved that there are knots that exist which are different from the unknot:

He did this by showing that two knot diagrams belong to the same type of knot if they can be related by a sequence of three Reidemeister moves:

Twist and untwist in either direction;

Move one loop completely over another;

Move a string completely under a crossing.

This was an extremely important result in knot theory. One important context in which these moves appear is to define knot invariants: by demonstrating that a property of a knot is unchanged after applying any of the Reidemeister moves, an invariant is defined.