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, Z_{G} = (1/16) (t_{1}^{8}+4t_{1}^{2}t_{2}^{3}+5t_{2}^{4}+2t_{4}^{2} +4t_{8}), and the coefficients of Z_{G}(x,1) = 1+x+4x^{2}+5x^{3}+8x^{4}+5x^{5 }+4x^{6}+x^{7}+x^{8} 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 Z_{G}(x,x,1) = 1+2x+12x^{2}+34x^{3}+87x^{4}+124x^{5 }+136x^{6}+72x^{7}+30x^{8} 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:
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 R_{1} = (1,9)(2)(3,4)(5)(6,7)(8) perpendicular to it. Hence, Z_{G} = (1/12)(t_{1}^{9}+4t_{1}^{3}t_{2}^{3}+3t_{1}t_{2}^{4}+2t_{3}^{3}+2t_{3}t_{6}), the coefficients of Z_{G}(x,1) = 1+2x+6x^{2}+12x^{3}+16x^{4}+16x^{5}+12x^{6}+6x^{7}+2x^{8}+x^{9} represent, respectively, the number of different symmetry choices of n-9 vertices for 9 £ n £ 18, and the coefficients of Z_{G}(x,x,1) = 1+4x+20x^{2}+76x^{3}+202x^{4}+388x^{5}+509x^{6}+448x^{7}+228x^{8 }+4x^{9} 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:
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. |