In 1926, Kurt Reidemeister (ride-a-my-stir) proved that if we have different representations (or projections) of the same knot, we can get one to look like the other using just three simple types of moves.
The Reidemeister Moves
|Take out (or put in) a simple twist in the knot|
|Add or remove two crossings (lay one strand over another)|
|Slide a strand from one side of a crossing to the other|
One thing to note in the above method is that even though by every Reidemeister move we make on the knot it changes the projection of the knot but it does not in any way change the knot represented by this projection. These Reidemeister moves can be performed on any knot given. Kurt Reidemeister proved that if the knot is represented by two distinct projections there has to be a way in which any combination of the Reidemeister moves can be performed on one projection to get to the other.