2.4 Derivation and classification of KLsThe first world we have derived is the linear world (or Lworld) that consists of KLs given by a general Conway symbol p (p ³ 1). For an odd p we obtain knots, and for an even p twocomponent links. All knots of the Lworld are periodical knots with graph symmetry group G = [2,p], and with knot symmetry group G' = [2, p]+, generated by a protation and 2rotation. The periods of every such knot are p and 2. For every knot of this family given by the Conway symbol p, the unknotting number is u(p) = [(p1)/ 2]. For each of them, the Alexander polynomial D(t) for knots, or reduced twovariable Alexander polynomial D(t, t) for 2component links, is a signalternating polynomial of order p+1, with all coefficients equal to 1. The linear world represents a subworld of the rational world. The rational world consists of alternating knots and twocomponent links. All its members are twobridge KLs, so all knots belonging to it are also known as twobridge knots. Thanks to the connection between rational KLs and continued fractions, all calculations with rational KLs are immensely simplified, including the recognition of KLs, verification of their equality, computation of unknotting and unlinking numbers, etc. This connection made it possible to construct specialized LinKnot functions for the computation and enumeration of particular classes of rational KLs with a desired property (e.g., to compute all achiral rational KLs, rational KLs with the unknotting (unlinking) number 1, etc.). The next world we consider is the stellar or prismatic world, abbreviated as Sworld. The name stellar, introduced by A. Caudron, comes from the corresponding graphs, and we also propose the more geometrical name prismatic. There we could distinguish the source links of the type 2,2,...,2 (Slinks), from the source links of the type 2,2,...,2+k (S^{+}links), where +k denotes a sequence of k pluses (k = 1,2). They are the base for derivation of all generating KLs, their corresponding infinite families, and all different particular KLs belonging to these families. For n £ 6, there occur some links already included in the rational world:
