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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 

n=8  (2,2)(2,2) 
n=9  (2,2) 1(2,2) 
n=10  (2,2,2)(2,2)  (2,2) 2(2,2) 
n=11  (2,2,2)~1(2,2)  (2,2) 3(2,2) 
n=12  (2,2,2)(2,2,2)  (2,2,2)~2(2,2)  (2,2) 4(2,2) 
(2,2,2,2)(2,2) 

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 

n=8  (2,2)(2,2) 
n=9  (~3 *,2)(2,2) 
n=10  (~3 *,2)(~3 *,2)  (2,2,2)(2,2) 
(~4 *,2)(2,2) 
(~3 *,~3 *)(2,2) 
n=11  (~5 *,2)(2,2)  (~3 *,2,2)(2,2) 
(~4 *,~3 *)(2,2)  (2,~3 *,2)(2,2) 
(~4 *,2)(~3 *,2)  (2,2,2)(~3 *,2) 
(~3 *,~3 *)(~3 *,2) 
n=12  (~4 *,2)(~4 *,2)  (~4 *,2,2)(2,2)  (2,2,2)(2,2,2)  (2,2,2,2)(2,2) 
(~3 *,~3 *)(~3 *,~3 * (2,~4 *,2)(2,2) 
(~6 *,2)(2,2)  (2,2,2)(~4 *,2) 
(~5 *,~3 *)(2,2)  (~3 *,~3 *,2)(2,2) 
(~5 *,2)(~3 *,2)  (~3 *,2,~3 *)(2,2) 
(~4 *,~4 *)(2,2)  (~3 *,2,2)(~3 *,2) 
(~4 *,~3 *)(~3 *,2)  (2,~3 *,2)(~3 *,2) 
(~4 *,2)(~3 *,~3 * (2,2,2)(~3 *,~3 *

By replacing every ~k * by k * we obtain from them all links of A1-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), (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 

n=10  (2,2),(2,2),2 
n=11  (2,2)1,(2,2),2 
n=12  (2,2),(2,2),(2,2)  (2,2),2,(2,2),2 
(2,2,2),(2,2),2  (2,2),(2,2),2,2 
(2,2)2,(2,2),2 
(2,2)1^2,(2,2),2 
(2,2)1,(2,2)1,2 

Finaly, for n £ 12, replacing in source A1-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 A2-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 A2-links, we replace digons in basic S-tangles of source A1-links by different basic S-tangles, etc. 

 

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