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

S' = (1,2,3,4,5,6,7,8,9,10)

and by reflection 

R = (1)(2,5)(3,4)(6)(7,10)(8,9). 
According to PET,  
ZG = (1/20)(t110+6t25+5t12t24+4t52+4t10), 
ZG(x,1) = 1+x+5x2+8x3 +16x4+16x5+16x6+8x7+5x8+x9+x10, 
ZG(x,x,1) = 1+2x+15x2+56x3+194x4+428x5+728x6 +800x7+636x8+272x9+78x10 
(10 £ n £ 20). For n = 11 we have the representative 10*a ({1}) generating 2 source links 10*2, 10*20; for n = 12 the representative 10*a.b ({1,2}, {1,3}, {1,6}, {1,7}, {1,9}) generating 3 source links 10*2.2, 10*2.20, 10*20.20. Taking for n = 11 the representative 10*2 (10*20) with P @ {(1)}, we could obtain for n £ 12 all links derived from 10*.  

To the basic polyhedron 10** corresponds graph symmetry group G = [2,2]+ of order 4, generated by two perpendicular 2-rotations  

S = (1,6)(2,7) (3,8)(4,9) (5,10) and 

S1 = (1,6)(2,5)(3,4)(7,10)(8,9). 
For it,  
ZG = (1/4)(t110+t12t24+2t25), 
ZG(x,1) = 1+3x+15x2 +32x3+60x4+66x5+60x6+32x7+15x8+3x9+x10, 
ZG(x,x,1) = 1+6x+53x2+248x3+874x4+2040x5
+3432x6+3872x7+2956x8+1296x9+288x10 

(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  

S = (1)(2,3,4,5) (6,7,8,9)(10), 

by reflection  

R = (1)(2,3)(4,5)(6,7)(8,9) (10) 

containing the rotation axis, and by reflection  

R1 = (1,10)(2,6)(3,7)(4, 8)(5,9) 
perpendicular to it, corresponds to the basic polyhedron 10***. For it,  

ZG = (1/16)(t110+2t12t42+3t12t24+2t16t22+6t25+2t2t42), and 
ZG(x,1) = 1+2x+8x2+13x3+25x4+25x5+25x6+13x7+8x8+2x9+x10 

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  

R = (1,5)(2,4)(3)(6,10)(7,9)(8)(11). 

For it,  

ZG = (1/2)(t111 +t13t24), 
ZG(x,1) = 1+7x+31x2+89x3+174x4+242x5+242x6+174x7+89x8+31x9+7x10+x11, 
ZG(x,x,1) = 1+14x+120x2+688x3+2700x4+7496x5+14944x6
+21312x7+21320x8+14256x9 +5728x10 +1088x11, 

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  

R = (1,11)(2)(3,4)(5)(6,7)(8)(9,10) and 

R1 = (1,10)(2,8) (3,6)(4,7) (5)(9,11) 

corresponds to the basic polyhedron 11**. For it,  

ZG = (1/4)(t111+2t1t25+t13t24), 
ZG(x,1) = 1+4x+18x2+47x3+92x4+126x5+126x6+92x7+47x8+18x9+4x10+x11,
ZG(x,x,1) = 1+8x+65x2+354x3+1370x4+3788x5
+7512x6+10736x7+10700x8+7208x9+2880x10+576x11. 

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  

R = (1)(2,3)(4)(5,6)(7,9) (8)(10,11) and 

R1 = (1)(2,10) (3,11)(4,8)(5,7)(6,9). 

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.

 

NEXT