In 1927, working with this diagrammatic form of knots, J. W. Alexander and G. B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below.
The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another.