12. Other basic polyhedra
The next member (2×5)^{*} of the infinite class
(2×k)^{*} is the basic polyhedron 10^{*} -
5-antiprism, with the graph symmetry group G = [2^{+},10]
of order 20, generated by rotational reflection
and by reflection To the basic polyhedron 10^{**} corresponds graph
symmetry group G = [2,2]^{+} of order 4, generated by two
perpendicular 2-rotations
(10 £ n £ 20). For n = 11 we have one representative 10^{**}a generating 2 source links 10^{**}2 and 10^{**}20; for n = 12 we have one asymmetrical representative 10^{**}a.b of the equivalence class consisting of eight 2-vertex choices, generating 4 source links 10^{**}2.2, 10^{**}2.20, 10^{**}20.2, 10^{**}20.20 and one symmetrical representative 10^{**}a:b of the equivalence class that consists of seven 2-vertex choices, generating 3 source links 10^{**}2:2, 10^{**}2:20, 10^{**}20:20. The graph symmetry group G = [2,4] of order 16,
generated by 4-rotation
by reflection containing the rotation axis, and by reflection
gives the number of different vertex bicolorings of n-10 vertices of 10^{***} for 10 £ n £ 20. Because the axis of 4-rotation contains two vertices of 10^{***}, we cannot use PET to obtain directly the number of source links derived from 10^{***}. For n = 11 we have two representatives of equivalence classes: 10^{***}a ({1}) giving 1 source link 10^{***}2, and 10^{***}.a ({2}) giving 2 source links 10^{***}.2, 10^{***}.20; for n = 12 we have three representatives of equivalence classes: 10^{***}a.b ({1,2}, {1,6}) generating 2 source links, 10^{***}2.2 and 10^{***}2.20, with P @ {(1)(2)}, 10^{***}.a:b ({2,4}, {1,10}, {2,6}, {2,8}) generating 2 source links 10^{***}.2:2 and 10^{***}.2:20 with P @ {(1,2)}, 10^{***}.a.b ({2,3}, {2,7}) generating 3 source links 10^{***}.2.2, 10^{***}.2.20, 10^{***}20.20 with P @ {(2,3)}. For n = 12, from 10^{***}2 we derive generating link 10^{***~}3 ^{*} , from 10^{***}.2 generating link 10^{***}.^{~}3 ^{*} , and from 10^{***}.20 generating link 10^{***~}3 ^{*}. To the basic polyhedron 11^{*} corresponds graph
symmetry group G = [1] of order 2, generated by reflection
For it, so we have the number of vertex choices and the number of source links derived from 11^{*} for 11 £ n £ 22. For n = 12 they are 7 vertex choices, and from each of them we derive 2 source links. The graph symmetry group G = [2] of order 4, generated
by two mutually perpendicular reflections
corresponds to the basic polyhedron 11^{**}. For
it,
For n = 12, from each of 4 vertex choices, we derive 2 source links. To the basic polyhedron 11^{***} corresponds the
graph symmetry group G = [2] generated by perpendicular reflections
Because the permutation representations of graph symmetry groups of 11^{**} and 11^{***} are isomorphic, we obtain the same enumeration result, and particular links could be translated from one basic polyhedron to the other by using that isomorphism. For n = 12 there exist 12 basic polyhedrons [15], given in 3D-form showing their symmetry, not always directly visible from their graphs or Schlegel diagrams. In the case of P-world for n £ 12, we will restrict the discussion of amphicheirality to the basic polyhedra and knots generated from them. |