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 |
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 |
|
  
|