1.5  Conway notation 

Having defined the basic notion of a tangle, we can explain the Conway notation. That symbolical notation for KLs was introduced by J.H. Conway in 1967 (Conway, 1970). The main advantage of the Conway symbols is the amount of important KL properties that are almost directly visible from the codes like: symmetry, recognition of the worlds (Caudron, 1982) to which particular KLs belong, the proof of the equality of rational KLs by using very simple calculation of corresponding continued fractions, and even derivation of some general conclusions. For example, all rational knots with an even number of crossings and with symmetrical (palindromic) Conway symbol are achiral, etc. Unfortunately, after the introduction of such fundamental concepts and the derivation of KLs with n £ 11 crossings, in the Conway paper KLs are divided into disjoint classes: knots, 2-component links, 3-component links etc., and the universal classification principle is unavoidably lost. In order to find it, we will reconsider the work by Alain Caudron (1982). Also, it is very surprising that the Conway notation, the only geometrical-topological notation that gives complete, interpretable and understandable information on KLs is still not universally accepted, and that in most knot theory books you will still find the classical notation of KLs. 

Maybe the main reason for this (and the main disadvantage of the Conway notation) is that most of KLs can be denoted in the Conway notation by several different symbols. In a similar way as with the classical notation, where every KL is given by its place in knot tables, we need to use some ßtandard" code, according to the notation introduced in the original Conway's paper (Conway, 1970) and in the papers and books following it (Caudron, 1982; Adams, 1994; Rolfsen, 1976). For example, the same link .2 can be denoted as :2, :.2, ::2, ::.2, or even as 6*2, 6*.2, 6*:.2, 6*::2, and 6*::.2. Also, we need to know particular symbols of basic polyhedra denoting their place in the list (database) of basic polyhedra. 

The LinKnot function fClassicToCon (webMathematica fClassicToCon) gives Conway symbol of a KL given in the classical notation. 

All the tangles are finite compositions of elementary tangles. The elementary tangles are 0, 1 and -1, where for alternating KLs 0 and 1 are sufficient. Tangles could be combined and modified by two operations : sum and product, leading from tangles a, b to the new tangles a+b, -a, a b, where -a is the image of a in NW-SE mirror line, a b = -a+b, and -a = a 0.