It is easy to notice that using Reidmeister moves W_{1} and W_{2} lessens the number of crossings. At first glance, it appears that there is a finite minimization algorithm for nonalternating projections: application of W_{1} move makes them proper, and of W_{2} decreases the number of crossings (in every step by 2). Unfortunately, things are not so simple and straightforward. In order to minimize a KL projection, sometimes it is necessary to enlarge the number of crossings and then reduce it by the application of W_{1} and W_{2}. In fact, the problem is that we don't know in which order Reidemeister moves should be applied: the sequence of moves producing the final result minimal projection. Examples of knots that can not be minimized without increasing the number of crossings are Nasty unknot ((1,3,1),1,1) (Adams, 1994), Goeritz's unknot ((1,1,3,1,1),1,1) (Goeritz, 1934), or Monster unknot 3 2 1 1 1 2 from R. Sharein's KnotPlot program. Usually we rather consider spherical then plane KL diagrams, thinking of the sphere as the plane compactified by the point at infinity. Without loos of generality, one can assume that the shadow of the KL does not contain this point. In this case, there appears one more "elementary isotopy" when some edge of the shadow passes through the infinity. This operation is called the infinity change (see, e.g., Manturov, 2004). By the infinity change operation we can obtain many plane diagrams from spherical diagrams. The smallest plane diagram that cannot be unknotted without increasing the number of crossings is Nasty unknot with n=7 crossings. However, considered as the diagram on a sphere, after the infinity change it can be reduced by the second Reidemeister move. An unknot (unlink) diagram on a sphere is called hard diagram if it has the following properties (Kauffman and Lambropoulou, 2006): 1) there are no simplifying type I or type II Reidemeister moves on the diagram; 2) there are no type III Reidemeister moves on the diagram. In a diagram there are type III
Reidemeister moves if it contains a triangle with one edge forming two
overcrossings or two undercrossings, so Monster unknot with n=10 crossings and Up to mirror images and flyping tangles in the diagrams, the smallest hard unknot and unlink diagrams have n=9 crossings. For n=9 there are two hard unknot diagrams and one hard unlink diagram; for n=10 we recognized six hard unknot diagrams and five hard unlink diagrams, etc.
