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7. Prismatic world

After R-world, we consider so-called "prismatic" or stellar world (S-world) [15]. There we could distinguish the basic links 2,2,¼,2 (S-links) and the direct products of basic links 2,2,¼,2+ (S×O-links), 2,2,¼,2++ (S×L-links). From them originate the corresponding subworlds of S-world, all generating links belonging to them, all different links and their infinite families, each of them defined by a generating link.  

For n £ 6, there occur some already discussed links: 2; 2+ = 3; 2++ = 22; 2,2 = 4; 2,2+ = 212; 2,2++ = 23 

For every even n (n ³ 6) we have (n/2)-component basic link 2,2,¼,2. Its graph is a n-gonal prism with colored lateral edges. Its graph symmetry group G = [2,n] is generated by n-rotation, vertical and horizontal plane reflection [4]. From this, we conclude that 2,2,¼,2 remains invariant after cyclic permutations of digons (rotations), where every such permutation is identified with its obverse (because of vertical reflection), or if all digons are obverted (horizontal reflection).

From a basic S-link 2,2,¼,2, we derive generating links replacing digons by ~R * -tangles: rational tangles not beginning by 1. The generating S~R* -links derived from 2,2,¼,2 according to the preceeding symmetry consideration for 6 £ n £ 12, are given in Table 2. All such links for fixed n ( n £ 12) could be obtained replacing in Table 2 every ~k * by k* 

Table 2  

n=6 2,2,2
n=7 ~*,2,2
n=8 ~*,2,2 2,2,2,2
~*,~*,2
n=9 ~*,2,2 ~*,2,2,2
~*,~*,2
~*,~*,~*
n=10 ~*,2,2 ~*,2,2,2
~*,~*,2 ~*,~*,2,2
~*,~*,2 ~*,2,~*,2
~*,~*,~*
n=11 ~*,2,2 ~*,2,2,2 ~*,2,2,2,2
~*,~*,2 ~*,~*,2,2
~*,~*,2 ~*,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
~*,~*,~* ~*,~*,~*,2
~*,~*,~* ~*,~*,2,~*
~*,~*,~*,~*

Now we may calculate the number of generating links and the number of all links derived from 2,2,¼,2 (Table 2). Every class, where ~n1* occurs k1 times, ~n2* occurs k2 times,¼, ~nm* occurs km times, consists of  

Õi = 1m  æ 
ç 
ç 
ç 
è 
~
d 
 
 * (ni)+ki-1 
ki
ö 
÷ 
÷ 
÷ 
ø 
= Õi = 1m  æ 
è 
fq+ki-1 
ki
ö 
ø 
 
generating links, where q=ni-2. Replacing each ~ni* by ni * , and every ~d * (ni) by d * (ni) (1 £ i £ m), we could enumerate all links of this class.  

The derivation of generating links from the direct products 2,2,¼,2+ and 2,2,¼,2++ completely follows the preceeding derivation from 2,2,¼, 2. To obtain from them all links for n fixed (n £ 12), we need to replace in the generating links obtained every ~k * by k * , and every ++ by 3,4,5,... pluses. Hence, for every n (6 £ n) we have the same number of prismatic (n+p)-crossing links with p pluses (p ³ 0).  

According to this, the structure of S-world is the following: all links of this world are derived from three infinite classes of basic links or their direct products (S-, S×O- and S×L-links) by R * -replacements, resulting in SR* -, SR * ×O- and SR * ×L-subworld.  

We could also analyze infinite families, representing the "vertical" structure of every world, where each family is obtained from certain generating link. For example, considering generating links from the first column of  Table 2, we obtain the infinite families: p,q,r (p ³ q ³ r ³ 2),  p1,q,r (p ³ 2, q ³ r ³ 2),  pq,r,s (p,q ³ 2, r ³ s ³ 2), p1,q1,r (p ³ q ³ 2, r ³ 2),  pq1,r,s (p,q ³ 2, r ³ s ³ 2),  p1q,r,s (p ³ q ³ 2, r ³ s ³ 2), p13,q,r (p ³ 2, q ³ r ³ 2),  pq,r1,s (p,q,r,s ³ 2),  p1,q1,r1 (p ³ q ³ r ³ 2), etc.  

Different combinatorial problems arise from here. For example, for every n fixed we could try to calculate the number of different classes occuring in the first column of Table 2 (beginning by series 1, 1, 2, 3, 4, 5, 7 for n £ 12), or the number of generating links occuring there, or even the number of all links derived from some generating link for every n fixed, etc. For example, the number of different classes mentioned is equal to the coefficient corresponding to qn-6 in [n-1,3], where by [n,r]  is denoted Gauss polynomial [36].

[n, r] =  (1-qn)¼(1-qn-r+1
(1-qr)¼(1-q)
.
 

All such problems belong to the theory of partitions with a given symmetry group (P-partitions). Let be given a permutation group P of  k objects and number n (n Î N, n ³ k). To every object ki (1 £ ki £ k) is assigned a natural number ni, where åi = 1k ni = n. Two partitions defined by assigned objects are equal iff there is a permutation from P transforming one to another; we need to find and enumerate different P-partitions. In some special cases, we could reduce problems of  P-partitions to the classical partition theory, but in general this is an open question. 

 

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