Reidemeister Moves

In the 1930s, Reidemeister first rigorously proved that knots exist which are distinct from the unknot. He did this by showing that all knot deformations can be reduced to a sequence of three types of "moves," called the (I) twist move, (II) poke move, and (III) slide move. These moves are most commonly called Reidemeister moves, although the term "equivalence moves" is sometimes also used (Aneziris 1999, p. 29).

Reidemeister's theorem guarantees that moves I, II, and III correspond to ambient isotopy (moves II and III alone correspond to regular isotopy). He then defined the concept of colorability, which is invariant under Reidemeister moves.

REFERENCES:

Aneziris, C. N. "The Equivalence Moves." Ch. 4 in The Mystery of Knots: Computer Programming for Knot Tabulation. Singapore: World Scientific, pp. 29-33, 1999.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.

Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, p. 16, 1991.

Reidemeister, K. "Knotten und Gruppen." Abh. Math. Sem. Univ. Hamburg 5, 7-23, 1927.