Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.

[1][2] Each move operates on a small region of the diagram and is one of three types:[3] No other part of the diagram is involved in the picture of a move, and a planar isotopy may distort the picture.

One important context in which the Reidemeister moves appear is in defining knot invariants.

[4] By demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves, an invariant is defined.

Many important invariants can be defined in this way, including the Jones polynomial.

The type III move is the only one which does not change the crossing number of the diagram.

The type I' move affects neither the framing of the link nor the writhe of the overall knot diagram.

[7] Furthermore, combined work of Östlund (2001), Manturov (2004), and Hagge (2006) shows that for every knot type there are a pair of knot diagrams so that every sequence of Reidemeister moves taking one to the other must use all three types of moves.

Coward & Lackenby (2014) proved the existence of an exponential tower upper bound (depending on crossing number) on the number of Reidemeister moves required to pass between two diagrams of the same link.

Lackenby (2015) proved the existence of a polynomial upper bound (depending on crossing number) on the number of Reidemeister moves required to change a diagram of the unknot to the standard unknot.