2.4  Derivation and classification of KLs 

The first world we have derived is the linear world (or L-world) 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 two-component links. All knots of the L-world are periodical knots with graph symmetry group G = [2,p], and with knot symmetry group G' = [2, p]+, generated by a p-rotation and 2-rotation. 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) = [(p-1)/ 2]. For each of them, the Alexander polynomial D(t) for knots, or reduced two-variable Alexander polynomial D(t, t) for 2-component links, is a sign-alternating 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 two-component links. All its members are two-bridge KLs, so all knots belonging to it are also known as two-bridge 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 S-world. 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 (S-links), 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:

2;   2+ = 3;   2++ = 2 2;   2,2 = 4;   2,2+ = 2 1 2;    2,2++ = 2 2 2.

 

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