At a first glance we can recognize a simple pattern: a regular periodic distribution of KLs in the family p, where knots (1-component links) are obtained for odd p, and 2-component links for even p (p = 2,3,...). In a similar way, if we consider KLs given by the general Conway symbol p q (e.g., the family derived from a figure-eight knot 4₁, expressed in Conway notation as 2 2), we will obtain links for p = q = 1 mod 2, and knots in all the other cases. For a more precise classification, every Conway symbol of a KL can be reduced by replacing all even numbers denoting chains of bigons by 2, all odd numbers greater or equal to 3 by 3, and all single vertices 1 remain unchanged. We call the KL obtained that way a generating link G. All KLs generated from it by adding an even number to each number that denotes a chain of bigons will make a subfamily generated from G. All the KL invariants will be strongly related to subfamilies and will represent some combination of numbers belonging to a general Conway symbol of a subfamily. In order to simplify, we will use the common term "family" for both families and subfamilies. 

We will first consider alternating KLs, and then non-alternating ones. One of the most powerful tools in our consideration will be symmetry. Symmetry could be visualized representing KLs by graphs, and trying to imagine them as 3-D illustrations- maps on a sphere. For every alternating KL we could distinguish the symmetry group G of its edge-bicolored graph where double edges are colored (bold) and where the relation "over-under" is disregarded, and its subgroup G' of index 2 (or antisymmetry subgroup G/G'), obtained by introducing the relation "over-under", that represents the actual symmetry of the KL. 

Symmetry groups of KLs, that is, groups of isometries that map a KL onto itself, are considered by B.Grünbaum and G.C.Shephard (1985). From 14 kinds of point groups in ³ (Coxeter and Moser, 1980), the 8 groups [q], [q]+, [2,q]+, [2+,2q], [2+,2q], [2,¥], [2,q] and [2,q+], where q is a positive integer, can occur as the symmetry groups of knots. In the case of links, all 14 kinds of point symmetry groups are permitted. Symmetries followed by a change of some bivalent property (e.g., sign +1 « -1, "black" « "white") are called antisymmetries , and corresponding groups are called antisymmetry groups.