2.5  Basic polyhedra and polyhedral KLs 

The next world is the polyhedral or P-world. The first problem here is the derivation of the basic polyhedra: 4-valent, at least 2-vertex connected graphs without bigons. 

This problem was solved for n £ 12 crossings by T.P. Kirkman (1885). The basic polyhedra are obtained by introducing triangular faces in KL diagrams, in order to eliminate all bigons. KL diagrams used for the derivation contain at most three bigons, all of them belonging to the same face. In this face we inscribe a triangle, with the vertices belonging to the face edges (e.g. in their midpoints), and each bigon contains a vertex of a triangle. The following table contains KL projections satisfying this necessary condition, Dowker codes of the derived basic polyhedra, and their list  

Table 5
 
 
n=3  4 6 2  6^* 
n=5  2 1 2  6 8| 2 10 4  8^* 
n=6  3 1 2  4 8 10 12 2 6  9^* 
6^*  6 8| 10 12| 2 4  9^* 
n=7  2 1 1 1 2'  4 8 10 12 2 14 6  10^*, 10^** 
.2  6 8| 10 12 14 2 4  10^**, 10^*** 
n=8  3 1 1 1 2  4 10 12 14 2 16 8 6  11^* 
2 1 2 1 2'  4 10 12 14| 8 2 16 6  11^** 
2 1 2 1 2'''  8 10 14| 2 16 4 6 12  11^* 
.3  6 8| 12 14 16|10 2 4  11^*, 11^** 
.21  6 8| 10 14 12 16 2 4  11^*, 11^** 
.2.20  6 8 14 12 4 16 2 10  11^*, 11^*** 
8^*  6 8 10 12 14 16 2 4  11^*, 11^** 
3#2 1 2  11^** 
n=9  3 1 2 1 2  4 12 10 16 14 2 18 6 8  12D 
2 1 3 1 2'  4 10 12 14 18 2 16 6 8  12D 
2 1 1 1 1 1 2  4 10 12 14 2 18 16 8 6  12A, 12B, 12F 
2 1 1 1 1 1 2'''  4 12 10 16 18 2 8 6 14  12B, 12F 
2 1,2 1,2 1  8 12 16| 2 18 4 10 6 14  12G 
.4  6 8| 12 14 16 18 2 4 10  12E 
.3 1  6 8| 10 14 16 18 2 4 12  12J, 12L 
.2 2  6 8| 16 14 12 18 2 4 10  12E 
.2 1 1  6 8| 12 14 18| 16 2 4 10  12B, 12H, 12I, 12J, 12K 
.3.20  8 10 12| 14 2 16 18 6 4  12D 
.2 1.2'  4 8 14 12 2 16 18 10 6  12B, 12F, 12H 
2:2:2  8 12 16| 2 14 4 18 6 10  12C 
.(2,2)  10 12| 14 18| 6 16 8 2 4  12I 
8^*2  8 10 12| 6 14 16 18 2 4  12B, 12F, 12G, 12H, 12I 
8^*20  6 8 10 16 14 18 4 2 12  12F, 12I, 12K 
9^*  6 16 14 12 4 2 18 10 8  12D, 12H, 12L 
2 1 2#1#3  12E 
6^*#3  12J 

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