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11. Basic polyhedra 8* and 9*

The next basic polyhedron 8* is 4-antiprism, with the graph symmetry group G = [2+,8] of order 16, generated by rotational reflection S' = (1,2,3,4,5,6,7,8) and reflection R = (1,3)(5,7)(4,8)(2)(6) containing its axis. The number of different symmetry choices of the vertices (i.e. vertex bicolorings of 8*) we could find using PET. In this case, ZG = (1/16) (t18+4t12t23+5t24+2t42 +4t8), and the coefficients of ZG(x,1) = 1+x+4x2+5x3+8x4+5x5 +4x6+x7+x8 represent, respectively the number of choices of n-8 vertices for 8 £ n £ 16. For 9 £ n £ 12, that choices are: {1}; {1,2}, {1,3}, {1,4}, {1,5}; {1,2,3}, {1,2,4}, {1,2,5}, {1,3,5}, {1,4,7}; {1,2,3,4}, {1,2,3,5}, {1,2,3,6}, {1,2,4,5}, {1,2,4,6}, {1,2,4,7}, {1,2,5, 6}, {1,3,5,7} corresponding, respectively, to the source links of the form 8*a; 8*a.b, 8*a:b, 8*a:., 8*a::b; 8*a.b.c, 8*a.b:c, 8*a.b:.c, 8*a:b:c, 8*a:.b:.c; 8*a.b.c.d, 8*a.b.c:d, 8*a.b.c:.d, 8*a.b:c.d, 8*a.b:c:d, 8*a:b.c:d, 8*a.b:.c.d, 8*a:b:c:d, given in Conway notation. By the coefficients of  ZG(x,x,1) = 1+2x+12x2+34x3+87x4+124x5 +136x6+72x7+30x8 is given the number of different source links derived from 8* for 8 £ n £ 16. All the vertex bicolorings obtained we could divide into equivalence classes, with regard to their symmetry groups, and then consider only their representatives. According to this, for n = 9 we have the representative 8*a giving 2 source links; for n = 10 the representative 8*a.b (8*a:b, 8*a:.b, 8*a::b) giving 3 source links; for n = 11 two representatives: 8*a.b.c (8*a:b:c, 8*a:.b:.c) giving 6 source links and 8*a.b:c (8*a.b:.c) giving 8 source links; for n = 12 five representatives: 8*a.b.c.d (8*a.b:c.d, 8*.a:b.c:d) giving 10 source links, 8*a.b.c:d (8*a.b:c:d) giving 16 source links, 8*a.b.c:.d giving 12 source links, 8*a:b:c:d giving 6 source links, and 8*a.b:.c.d giving 7 source links, where the other members of equivalence classes are given in parentheses. The list of source links derived from that representatives is given in Table 9:  

Table 9  
 

n=9  8*2  n=11  8*2.2.2  8*2.2:.2 
8*20  8*2.2.20  8*2.2:.20 
8*2.20.2  8*2.20:.2 
n=10  8*2.2  8*2.20.20  8*20.2:.2 
8*2.20  8*20.2.20  8*2.20:.20 
8*20.20  8*20.20.20  8*20.2:.20 
8*20.20:.2 
8*20.20:.20 
n=12  8*2.2.2.2  8*2.2.2:2  8*2.2.2:.2  8*2:2:2:2  8*2.2:.2.2 
8*2.2.2.20  8*2.2.2:20  8*2.2.2:.20  8*2:2:2:20  8*2.2:.2.20 
8*2.2.20.2  8*2.2.20:2  8*2.2.20:.2  8*2:2:20:20  8*2.2:.20.20 
8*2.2.20.20  8*2.20.2:2  8*2.20.2:.2  8*2:20:2:20  8*2.20:.2.20 
8*2.20.2.20  8*20.2.2:20  8*2.2.20:.20  8*2:20:20:20  8*2.20:.20.2 
8*2.20.20.2  8*2.2.20:20  8*2.20.2:.20  8*20:20:20:20  8*2.20:.20.20 
8*20.2.2.20  8*2.20.2:20  8*20.2.20:.2  8*20.20:.20.20 
8*2.20.20.20  8*20.2.2:20  8*20.20.2:.2 
8*20.2.20.20  8*2.20.20:2  8*20.2.20:.20 
8*20.20.20.20  8*20.2.20:2  8*20.20.2:.20 
8*20.20.2:2  8*20.20.20:.2 
8*2.20.20:20  8*20.20.20:.20 
8*20.2.20:20 
8*20.20.2:20 
8*20.20.20:2 
8*20.20.20.20 

After that, the links of PR* -subworld derived from 8* we obtain replacing digons in the source links by R*-tangles. Using the symmetry equivalents, we could reduce again a complete derivation to that from the corresponding representatives. For n = 9 we have the representative 8*2 (8*20) with P @ {(1)}; for n = 10 two representatives: 8*2.2 (8*20.20) with P @ {(1,2)} and 8*2.20 with P @ {(1)(2)}; for n = 11 two representatives: 8*2.2.2 (8*2.20.2, 8*20.2.20, 8*20.20.20) with P @ {(1,3)(2)}, and 8*2.2.20 (8*2.20.20, 8*2.20.20, and all source links derived from 8*2.2:.2) with P @ {(1)(2)(3)}, where the other members of equivalence classes are given in parentheses. That permutation groups P are already considered in such derivation from 6*, so it will be not repeated.  

For every even n ³ 8 (n = 2k), there is k-antiprism - the basic polyhedron of the form n* = (2×k)* [14], with the graph symmetry group G = [2+,n] of order 2n generated by rotational reflection S' and reflection R, so all the results obtained for 8* could be generalized.  

The graph symmetry group G = [2,3] of order 12, corresponding to the basic polyhedron 9* is generated by 3-rotation S = (1,4,7)(2,5,8)(3,6,9) and by two reflections, R = (1)(2,8)(3,6)(4,7)(5)(9) containing the rotation axis and R1 = (1,9)(2)(3,4)(5)(6,7)(8) perpendicular to it. Hence, ZG = (1/12)(t19+4t13t23+3t1t24+2t33+2t3t6), the coefficients of  ZG(x,1) = 1+2x+6x2+12x3+16x4+16x5+12x6+6x7+2x8+x9 represent, respectively, the number of different symmetry choices of  n-9 vertices for 9 £ n £ 18, and the coefficients of  ZG(x,x,1) = 1+4x+20x2+76x3+202x4+388x5+509x6+448x7+228x8 +4x9 the number of source links derived from 9* for 9 £ n £ 18. For n £ 12, that vertex choices are divided into symmetry equivalence classes and given by their representatives. For n = 10 we have one representative 9*a ({1}, {2}) generating 2 source links; for n = 11 two representatives: 9*a.b ({1,2}, ({1,5}) generating 4 source links, 9*a:b ({1,3}, {1,4}, {1,9}, {2,5}) generating 3 source links; for n = 12 three representatives: 9*a.b.c ({1,2,3}, {1,2,4}, {1,2,8}, {1,2,9}, {1,4,6}, {1,5,9}) generating 6 source links, 9*a.b:.c ({1,2,5}, {1,2,6}, {1,3,4}, {1,4,5}) generating 8 source links, 9*a:.b:.c ({1,4,7}, {2,5,8}) generating 4 source links. The list of the source links derived from that representatives is given in Table 10: 

Table 10 
 

n=10  9*.2  n=12  9*2.2.2  9*2.2:.2  9*2:.2:.2 
9*.20  9*2.2.20  9*2.2:.20  9*2:.2:.20 
9*2.20.2  9*2.20:.2  9*2:.20:.20 
n=11  9*2.2  9*2:2  9*2.20.20  9*20.2:.2  9*20:.20:.20 
9*2.20  9*2:20  9*20.2.20  9*2.20:.20 
9*20.2  9*20:20  9*20.20.20  9*20.2:.20 
9*20.20  9*20.20:.2 
9*20.20:.20 

The links of  PR* -subworld derived from 9* we obtain replacing digons in the source links by R*-tangles. Using the symmetry equivalents, we reduce a complete derivation to that from the corresponding representatives. For n = 10 we have the representative 9*2 (9*20) with P @ {(1)}; for n = 11 two representatives: 9*2:2 (9*20:20) with P @ {(1,2)}, 9*2.2 (9*2:20, and all source links derived from 9*2.2) with P @ {(1)(2)}. Their permutation groups P are considered before.  

 

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