## The Reidemeister Moves

An important consideration in finding an isotopy invariant is how a knot can be bent and twisted into another. Such bending and twisting is known as an ambient iosotopy, or just as an isotopy. Two or more knots that can be turned into each other through an ambient isotopy are considered ambiently isotopic, or isotopic in respect to each other. Finally, any group of knots that are isotopic in respect to each other belong to the same isotopic class. Kurt Reidemeister was able to prove that any ambient isotopy can be performed with only three types of moves. These moves became known as the Reidemeister moves.

Although not a true Reidemeister move, there is a move that allows strands to be changed in shape and length without affecting any crossings. This move is usually written as R0

The first Reidemeister move, shown below and denoted with R1, simply adds or removes one crossing through a simple loop.

The second Reidemeister move, denoted with R2, adds or removes two crossings simultaneously from a knot.

The third Reidemeister move, denoted with R3, allows a strand to be moved to the other side of a crossing.

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