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.