Oriented Reidemeister moves

An oriented Reidemeister move is one of the following local replacements of a regular oriented link diagram (Regular oriented link diagrams), performed inside a small closed disk while the diagram outside the disk is left unchanged, and understood up to planar isotopy outside the disk (Planar isotopy of link diagrams). All sign and orientation variants of each local picture are included.

(R1) Kink creation and deletion. A single strand is replaced by a small kink: two arcs crossing once and otherwise disjoint from the strand, with the over/under datum of the new crossing as in the picture. The move is allowed in both directions (create or delete the kink) and with both signs of the kink, the sign being the sign of the crossing created. The orientations of the two strands of the kink are the two possible consistent orientations through the crossing, and all of them occur in the move list.

(R2) Slide of two strands. Two local arcs belonging to two strands are replaced so that the number of crossings between those strands changes by two of opposite signs: the standard picture slides one strand over or under the other, creating or deleting the pair of crossings, with the two crossings being the two possible signs in either order. All four orientation patterns of the two strands at the two crossings are included.

(R3) Sliding a strand past a crossing. One strand is moved across an existing crossing of two other strands, so that the crossing of the two other strands passes from one side of the moving strand to the other; the three crossings of the local picture keep their signs and the over/under data are preserved. The three strands must have a consistent strict height order: one is above both others, one below both, and the remaining strand is between them. The cyclic over/under pattern is excluded. Every orientation pattern and all six height orders of this standard local picture are included.

A local R2 or R3 picture is braid-like when, after a local coordinate change and deformation, all participating strands run in the same positive longitudinal direction. This is a local condition and does not require a globally chosen braid axis. R1 is kept as a separate kink move.

The strands keep their orientations throughout every move. The union of the three move types, together with planar isotopy, generates the equivalence relation on diagrams used in the Reidemeister equivalence theorem below; the moves are sometimes called R1, R2, R3 as above. Since every move changes the diagram only inside a disk, it has a type and a finite list of local orientation/sign variants, and each variant is realized by an ambient isotopy as the next items show. The complete list of variants is required because the Markov and Alexander constructions below act on oriented diagrams and count signed crossings.