8. Arborescent world

The next world is arborescent (or A-world) [15]. Its members are the multiple combinations of links belonging to the preceeding worlds. Because A-links with pluses could be directly obtained from the corresponding A-links without them, we will restrict our attention to the others. If the basic and generating links of S-world are treated as A₀-level, the source links defining the first level (A₁-level) of A-world are obtained replacing digons in basic S-links 2,2,¼,2 or every first digon in ^~R ^*-parts of generating S^~R ^*-links by basic S-tangles (2,2,¼,2) (including (2,2)). 

If only one such digon in a basic S-link or in generating S^~R ^*-link is replaced, for n £ 12 we have 16 source A₁-links of the form X,2,¼,2, where X is the digon or ^~R ^*-part after this substitution. According to the relationship X,2,¼,2 = X(2,¼,2), they could be written in a symmetrical form, as the rational generating links with the first and the last digon replaced by a basic S-tangle. Their list for n £ 12 is: 

Table 3 

The further derivation of generating A₁-links from the source links is conditioned by their symmetry. Therefore, among the source A₁-links from the first column of  Table 3 we distinguish links with the graph symmetry group G = [2⁺,4] ((2,2)(2,2) and (2,2,2)(2,2,2)), and links with the group G = [2] ((2,2,2)(2,2), (2,2,2,2)(2,2), etc.). The derivation of A₁-generating links from them is given in the following table: 

Table 4 

The effect of 2-antirotation existing in (2,2)(2,2) is preserved in all symmetrical knots derived from it. Hence, for n £ 12 we have the amphicheiral [22-29] knots (3,2)(3,2), (21,2)(21,2), (2²,2)(2²,2), (21²,2)(21²,2), (3,3)(3,3), (21, 21)(21,21), and all of them are non-invertible. Also, there are 3-component links (2,2)(2,2) (or 8₄³, known from the work [17] as amphicheiral), (4,2)(4,2) and (31,2)(31,2), that are amphicheiral as well. Hence, (2,2)(2,2) generates infinite series of amphicheiral knots and 3-component links (e.g. knots of the form (p,q)(p,q), p ¹ q (mod 2), and 3-component links of the form (p,q)(p,q), p = q (mod 2), etc.). 

From the generating links without pluses we directly obtain analogous A₁-generating links with pluses, derived from source links (2,2+)(2,2), (2,2++)(2,2), (2,2+) (2,2+), (2,2++)(2,2+), (2,2++)(2,2++), etc. From every symmetrical generating A₁-link from Table 4 we obtain 5 such links of each class, and from every asymmetrical link 8 of them. After that, by replacing every ^~k ^* by k ^* and every ++ by 3,4,5,... pluses, and taking care about symmetry, we obtain from them all links of A₁-level with pluses. Among them, for n £ 12, there are two amphicheiral knots: (3,2+)(3,2+) and (21,2+)(21,2+). 

The further derivation of generating and other links from the source links (2,2) 1 (2,2) and (2,2,2)1(2,2), belonging to the second column of Table 3, completely follows the derivation from (2,2)(2,2) and (2,2,2)(2,2), respectively (Table 4). 

From the source link (2,2) 2(2,2), for n = 11 we obtain generating links (2,2) 3 (2,2), (3 ^*,2)^~2(2,2), and for n = 12 are derived generating links (2,2) 4(2,2), (^~3 ^*,2) 2 (2,^~3 ^*), (^~4 ^*,2)^~2(2,2), (^~3 ^*,^~3 ^*)^~2(2,2), (3 ^*,2)^~2(^~3 ^*,2), (3 ^*,2)^~3 ^*(2,2). The corresponding generating A₁-links with pluses and all the corresponding A₁-links could be obtained from them in the same way as before. 

The other source A₁-links of the arborescent world are obtained replacing k digons in a basic S-link or every first digon in ^~R ^*-part of a generating SR ^*-link by basic S-tangles (2,2,¼,2) (including (2,2)), where k (k ³ 2) such digons are replaced (Table 5). As well as before, the obtained source A₁-links could be also writen in another form (e.g. (2,2),(2,2),2=((2,2),2)(2,2); (2,2)1,(2,2),2=((2,2),2)1 (2,2), etc.) 

Table 5