8. Arborescent worldThe 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 A_{0}-level, the source links defining the first level (A_{1}-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 A_{1}-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:
The further derivation of generating A_{1}-links from the source links is conditioned by their symmetry. Therefore, among the source A_{1}-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 A_{1}-generating links from them is given in the following table:
By replacing every ^{~}k ^{*} by k ^{*} we obtain from them all links of A_{1}-level for n £ 12. Using the combinatorial formula for the number of prismatic links, we could calculate their number. 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), (2^{2},2)(2^{2},2), (21^{2},2)(21^{2},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 8_{4}^{3}, 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 A_{1}-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 A_{1}-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 A_{1}-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 A_{1}-links with pluses and all the corresponding A_{1}-links could be obtained from them in the same way as before. The other source A_{1}-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 A_{1}-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.)
Finaly, for n £ 12, replacing in source A_{1}-link (2,2),(2,2),2 a digon in a S-tangle (2,2) by the same tangle (2,2), we obtain the first source link ((2,2),2),(2,2),2 of the next A_{2}-level. It could be written in the symmetrical form as ((2,2),2)((2,2),2). In the further procedure (for n ³ 12), to obtain all source A_{2}-links, we replace digons in basic S-tangles of source A_{1}-links by different basic S-tangles, etc. |