## 1.3  KL diagrams

In order to obtain a KL shadow, we project a 3-D link L into the plane Â2. To avoid the loss of the information on three-dimensionality of L, we need the relation "over-under". We can draw KL shadows in such a way that one arc passes over or under another arc at each crossing. The pictures obtained that way are link diagrams or link projections. The notion of a proper shadow can be directly transferred to a proper diagram (or reduced diagram)- a KL diagram without loops.

Let us now take an arbitrary set of points P1, P2,..., Pc belonging to shadows of distinct components on a diagram L' of a link L, and move them each along the corresponding component shadow. If every point Pi (i = 1,...,c), travelling around the component in a fixed direction meets crossings that alternate between over and under, the diagram L' is called an alternating diagram. A link L that possesses at least one alternating diagram is called an alternating link. Otherwise, it is a non-alternating link. At the beginning of knot theory, all knots were thought to be alternating. The simplest non-alternating knots occur among 8-crossing knots, and simplest non-alternating link is the 6-crossing link 633 (2,2,-2). To every KL corresponds an infinite number of diagrams representing it. In the case of alternating KLs, we are searching among them for at least one alternating diagram. It is by no means trivial to prove that we can even find it.

Among all corresponding diagrams of a link L we can distinguish those with a minimum number of crossings. If n is the least number of crossings in any projection of L, it is called minimum crossing number (or just crossing number) of L.

Regular (generic) KL-shadows (without splitting points) are already defined (see also, e.g., Murasugi, 1996, page 26; Sosinsky, 2002 , pages 38-39). We will deal only with regular
proper KL diagrams (regular KL diagrams without loops), so if there is no explicit remark that some KL diagram is non-proper, the term ``diagram of a KL'' means ``a regular proper diagram''.

Probably the most important property of alternating diagrams is that they are minimal: each alternating diagram of a link L is reduced to a minimum number of crossings. In 1986, L.Kauffman, K.Murasugi and M.Thistlethwaite, using properties of the Jones and Kauffman polynomial, independently proved the famous First Tait Conjecture that an alternating KL in a reduced alternating projection of n crossings has crossing number n (Kauffman, 1987b; Murasugi, 1987a,b; Thistlethwaite, 1987, 1988 ). This theorem is now known as Kauffman-Murasugi Theorem.

In general, it is very difficult to determine the crossing number of a given KL. If we have a non-alternating KL projection with k crossings, is there a hope of showing that it can be drawn with fewer than k crossings? How to find the crossing number n and prove that it is really the minimum number of crossings for a given link L?