## 1.4  Reidemeister moves

The next step in derivation of all KLs with a given number of crossings is finding all different (non-isomorphic) minimal projections of a given KL, i.e., all its different projections with the number of vertices equal to the crossing number. In the case of alternating KLs it is enough to find all proper non-isomorphic alternating projections of a given KL with a fixed number of crossings. Non-alternating KLs can be derived from them by crossing changes.

KLs are usually given by their non-minimal projections, so it is necessary to minimize them. There is a finite algorithm guaranteeing such a minimization, based on Haken-Hemion method (Haken, 1961; Hemion, 1979). However, it is almost impossible to implement because of its complexity. In spite of this, there are two computer programs attempting to produce best possible minimizations. The first of them is part of Knotscape, the program knotfind.c based on a heuristic algorithm that works only with knots. The other is the reduction program that works with links as well, written by M. Ochiai and N. Imafuji, and included in the Mathematica-based program Knot 2000 (K2K) as the basis of the function ReductionKnotLink (webMathematica ReductionKnotLink). Both of them are not error-free and sometimes produce wrong results, but most of their reductions are the optimal ones. Before a more detailed discussion of that problem, we will describe the elementary steps of a reduction process: Reidemeister moves. Till now we had only static images: KL shadows and projections. Now we are ready to start with a movie, in which ambient isotopy plays the main role. All KLs will be represented as polygonal KLs, and all the moves that consist of a finite series of elementary isotopies will be expressed as finite sequences of Reidemeister moves (or changes). The move W0 was already introduced as an elementary isotopy. For a polygonal link an planar isotopy W0 is achieved either by subdividing an edge AB by the vertex C, or contracting AC and CB. An ambient isotopy for a polygonal KL is a finite sequence of elementary isotopies. The other three moves, W1, W2, and W3 are illustrated in the following figure. We represented Reidemeister moves as polygonal moves, and the piecewise-linear and the smooth knot theory give the same classification of KLs.

In the case of alternating KLs Reidemeister moves are unnecessary for the transformation of one minimal projection to another minimal projection. Instead of them, we can use flypes, moves introduced by P.G. Tait. They are well known from his Flyping Conjecture (1876/77), that after the proof provided by W.Menasco and M.Thistlethwaite in 1990 become the Tait's
Flyping Theorem
(Menasco and Thistlethwaite, 1991, 1993). A natural way of expressing a flype as a sequence of Reidemeister moves is still unknown.