Reidemeister Moves

We list the Reidemeister moves in Fig.

and would like to leave details to other books in knot theory, and recall the most fundamental theorem of Reidemeister we rely on.

**Figure:** reidmoves
$\begin{figure}\begin{center} \mbox{ \epsfxsize =4in \epsfbox{reidmoves.eps}} \end{center}\end{figure}$

**Figure:** Moves with a height function specified
$\begin{figure}\begin{center} \mbox{ \epsfxsize =2.5in \epsfbox{FrYet.eps}} \end{center}\end{figure}$

Remark 2.2 There are variations of Reidemeister moves in various situations and set-ups. Here we mention three variations.

The Jones polynomial was first discovered through braid group representations [Jones87]. Two theorems play key roles (see, for example, [Kau91,Kama02]): Alexander's theorem says that any knot or link can be represented as a closed braid, and Markov's theorem says how closed braid forms of the same knot are related (related by congugations and Markov de-stabilizations). Thus the relations of the braid groups and Markov (de-)stabilizations play the role of Reidemeister moves.
In defining the Jones polynomial from an operator approach [Kau91], a height function is fixed on the plane, and local maxima and minima of knot diagrams plays an important role (a pairing and a copairing are assigned to them). Then moves involving these become necessary to take into considerations. These additional moves, called Freyd-Yetter moves and depicted in Fig. , are considered also in formulating category formed by knot diagrams [FY89].
For virtual knots, Reidemeister moves are symbolically interpreted in terms of Gauss codes, then reinterpreted in terms of virtual knot diagrams, see [Kau99], for example. Those moves involving virtual crossings are depicted in Fig. .
Another variation is to assign parenthesis structures on arcs of the diagrams, called non-associative tangles [BN97]. These were used for the study of finite type invariants of knots.